A first analysis of the sources
You've done a huge work by gathering all these different sources!
Here a short first feedback:
As you mentioned the 5th line seemingly only appears in the book of as-Suyuti, and it is a matter of discussion whether this line is part of it or was added (Mualla 'Ali al-Qari mentioned a slightly different wording of this line and an additional line or verse, while az-Zurqani discussed a second version of the 8th line which he claimed is not eloquent enough).
Note: As the sources of the text are provided in the original post I didn't add a link to the sources of my translated text quotes out of these.
Basically from my superficial first go through all the provided reports are narrated on the authority of the sahabi Khoraim ibn Aws at-Taaiy خريم بن أوس الطائي who actually converted to Islam after the battle of Tabook. Al-Hakim claimed that the narrators were 'Araab (Bedouins) and they can't be liars in this regard:
In the following I'll be translating from Arabic language as these translations are of my own take them with the necessary care!
هذا حديث تفرد به رواته الأعراب عن آبائهم، وأمثالهم من الرواة لا يضعون
This hadith is unique to its narrators from Arabs of the desert from their fathers, and narrators of this kind do not (tend to) fabricate (hadith).
but a-Dhahabi commented in his Siyar still they are unknown:
قال الحاكم: رواته أعراب، ومثلهم لا يضعفون.
قلت: ولكنهم لا يعرفون.
Al-Hakim said: it's narrators are Arabs of the desert and narrators of this kind can't be declared weak.
I say: but they are unknown!
and so did al-Haythami in his Majma' az-Zawaaid referring to at-Tabarani's sanad in al-Mu'ajam al-Kabir by saying:
رواه الطبراني، وفيه من لم أعرفهم.
It was narrated by at-Tabarani and in its chain are (people which are) unknown to me.
As-Suyuti in his al-La'ali' al-Masno'ah اللآلىء المصنوعة في الأحاديث الموضوعة confirmed that the hadith claiming that this poem was attributed to Hassan ibn Thabit is a fabrication and confirmed that this poem is of al-'Abbbas, saying (See here at the bottom and at the beginning of the following page):
مَوْضُوع: وَضعه بعضُ الْقصاص وهناد لَا يوثق بِهِ وَلَعَلَّه من وضع شَيْخه أَو شيخ شَيْخه والأبيات للْعَبَّاس بِلَا خلاف
fabricated: This was fabricated by some story tellers and Hannaad is not trustworthy. Perhaps it was one of his sheikh's fabrication or the sheikh of his sheikh. And the verses are for al-Abbas without any disagreement.
It should also be noted that the length of the report has differences for example the reports of abu Nu'aym and that of al-Khateeb al-Baghdadi mention an addition which at-Tabarani and al-Bayhaqi (via al-Hakim) for example mentioned as a stand alone hadith right after this report.
Note: Evil said this could be a kind of work around to make an unknown or a person who had only reported/narrated one hadith to a person who at least reported two or three ahadith or totally the opposite. But I don't know for sure and I don't want to spread anything false or malicious.
Some scholars like ibn 'Asakir and ibn Kathir also quoted that this poem was also attributed to Hassaan ibn Thabit, both agreed that this is a wrong claim.
Summary and qualification
At this point we could say that the above qualifications mean that a-Dhahabi would regard the hadith as complied by al-Hakim as da'if. While al-Hakim accepts unknown narrators as mentioned in: Is there a scale or classification for scholars and their qualification of hadith narrators?
and therefore would regard the hadith as sahih.
The same may apply to al-Haythami's qualification of the hadith from at-Tabarni's al-Kabir, however I don't know al-Haythami's position on the topic of unknown narrators and narrators who have narrated only one or two hadith.
However many scholars regard him as rather lenient for example in qualification of narrators as those of both sahih or trustworthy etc. and for the fact that he may regard a narrator who is matrook or a fabricator only da'if and for his acceptance or agreement with the qualification of ibn Hebban on a rawy ibn Hebban was not sure about, meaning an unknown (which is also mentioned in above linked post where ibn Hebban hold the same position as al-Hakim in his method of qualification).
Having that said the hadith has at least a strong tendency to be weak.
The following deeper analysis will only show that many trustworthy scholars have reported the hadith from these narrators who at least would have a level of doubt.
A bit off-topic:
Imam az-Zurqani in his commentary on al-Mawahib al-Laduniya شرح الزرقاني على المواهب اللدنية بالمنح المحمدية commented on the Arabic language of this poem and in cases referring to different narrations came to the conclusion that some of the lines of some versions of the verses can't be from such a fluent and eloquent speaker as al-'Abbas (See here).
The hadith has been referred to by some other scholars like ibn al-Atheer in his an-Nihaya fi gharib al-Hadith النهاية في غريب الحديث والأثر in which he explained some words quoting a verse of the poem (see here, here and here).
There also seem to be an intersection in the content and a permutation of the order between this poem and one attributed to al-'Abbas ibn Mirdas العباس بن مرداس as claimed by a-Diwan and other pages sharing Arabic poetry such as adab:
وَأَنتَ لَمّا وُلِدتَ أَشرَقَتِ ال ...-... أَرضُ وَضاءَت بِنورِكَ الأُفُقُ
مِن قَبلِها طِبتَ في الظِلالِ وَفي ...-... مُستَودَعٍ حَيثُ يُخصَفُ الوَرَقُ
Checking the sanad of the books which provide one
First we should agree that the following books or authors don't mention a sanad (some don't even mention a source of the quote):
- al-Khasais al-Kubra الخصائص الكبرى of as-Suyuti only refers to al-Bayhaqi and at-Tabarani's books.
- Majma' az-Zawaaid مجمع الزوائد ومنبع الفوائد of al-Haythami refering to at-Tabarani's book.
- al-Isti'aab الاستيعاب في معرفة الأصحاب of ibn 'abd al-Barr He mentioned that the same hadith was reported once on the authorithy of Khoraim ibn Aws and once by his brother Jarir ibn Aws and he called the hadith as lengthy.
- Neither in a-Shifa الشفا بتعريف حقوق المصطفى of al-Qadi 'Iyad nor the version with the side notes of a-Shumni on it, or in any of its commentaries like that of Mualla 'Ali al-Qari the hadith or a source is mentioned as Qadi 'Iyad only quoted the poem of al-'Abbas.
- Zaad al-Ma'ad زاد المعاد في هدي خير العباد of ibn al-Qayyim.
Further later scholars as a-Dhahabi and ibn Kathir quoted the sanad based on a different source, so the narrator chain is not connected between them and this source.
Due to the fact that the earliest sources in which this hadith was mentioned are the books of ibn Qutaybah a-Dinawri (213 a.H. -276 a.H.), abu Bakr a-Shafi'i (260 a.H. - 354 a.H.) (ibn Asakir reported via him at least one chain) and at-Tabarani (260 a.H. -360 a.H.) we'll start the analysis from these chains:
Discussion of some issues among the "source narrators"
One first remark all these three "old" sources quote a narrator which was referred to as:
عم أبي زحر بن حصن
The uncle of my father Zahr ibn Hisn.
ibn 'Asakir, abu Nu'aim al-Asbahani, ibn Kathir, al-Bayhaqi and al-Hakim followed their example.
In the copy of Maktaba of al-Bayhaqi's a Dalail they have mistakenly called him
عم أبي زخر بن حصن
The uncle of my father Zakhr ibn Hisn.
while ibn Kathir reported the hadith via al-Bahyaqi with the correct reference in his Seerah an-Nabawiyyah.
while later sources use a name instead the reference uncle!
Al-Khateeb al-Baghdadi calls him:
عمر أبو زحر بن حصن
'Umar abu Zahr ibn Hisn
while the editors of Siyar 'Alaam an-Nubala' pointed at many typos in the source manuscripts in the Tareekh of a-Dhahabi a huge mistake was performed by naming him:
زحر بن حسن
Zahr ibn Hassan.
While he named him in his Siyar:
زحر بن حصن
Zahr ibn Hisn
All this seems to show a huge confusion about an unknown rawy (narrator). However it seems the rawy is Zahr ibn Hisn زحر بن حصن and due to ignorance some people copied the name or falsely "corrected" it while copying the book. And this Zahr ibn Hisn was the uncle of the father of his narrator Zakariya ibn Yahya زكريا بن يحيى also known as abu as-Sukayn أبو السكين as is best explained by his full name as mentioned by abu Bakr a-Shafi'i and also to some extent by al-Bukhari in his a-Tarikh al-Kabir (See #1486 here).
In some books of Jarh wa t'aadil we may find the following information about a man referred to as:
زحر بن حصن
Zahr ibn Hisn
All that both abu Hathim ar-Razi and al-Bukhari agree about him is that he narrated from his grand father Humayd ibn Munhib and his narrator Zakariya ibn Yahya ibn ‘Umar ibn Hisn at-Taa’i also known as abu as-Sukayn. However ibn Hebban said he narrated from his uncle and father and gave him the kunya abu al-Faraj and said that he died 104 a.H..
As for his grand father حميد بن منهب Humayd ibn Munhib ibn 'Abd al-Barr denied that he was sahabi and said he narrated from 'Ali and 'Uthman. The hadith database pretends that he also narrated from the sahabah: the mother of believers 'Aishah, Khoraim -the narrator of our hadith who is his grand father-, Mua'awiya ibn abi Sufyan and 'Urwah ibn Mudarris at-Taa'i. None of the authors of the 6 books nor imam Malik reported via him.
So basically we have discussed both narrators which create an issue here none of these both is mentioned in any of the 6-7 main sunnah references among sunni scholars nor did they report any hadith on the authorithy of the sahabi reporting the hadith wed iscuss here `horaim ibn Aws at-Taa'i خريم بن أوس.
This again shows how lenient al-Hakim was as "the origin" of the report itself can't be claimed to be as of one of both sahihs unless one regards all sahabah as trustworthy -which is the view among sunni scholars- and accepts rather unknown narrators as trustworthy even if they are not narrators from both sahihs nor do scholars know much of them except with this and some more narrations.
Before moving further let me list and if not yet done introduce the four narrators on whom our three oldest sources agree and as at least indicated above all sources OP has shared would agree on:
- Khoraim ibn Aws ibn Haritah ibn Laam at-Taai' خريم بن أوس who witnessed this as he just converted to Islam. And he is the grand father of
- Humayd ibn Munhib حميد بن منهب, who already was introduced and he is the grand father of
- Zahr ibn Hisn زحر بن حصن, less lenient scholars as a-Dhahabi regard him as unknown which would be a "defect" in the narrator chain for any scholar who is hard liner or moderate in the subject of al-Jarh wa a-Ta'adeel. He is the -paternal- uncle of
- Zakariya ibn Yahya ibn 'Umar ibn Humayd ibn Harith ibn Khoraim ibn 'Aws ibn Harithah ثنا زكريا بن يحيى بن عمر بن حصن بن حميد بن منهب بن حارث بن خريم بن أوس بن حارثة (the full name listing his fathers fathers till our hadith source of the scholar known as abu as-Sukayn أبو السكين) who appears in one hadith as the teacher of al-Bukhari here. He died 235 a.H. according to Mughlatay (see here). Al-Hakim himself said he is reporting ahadith with mistakes, ibn Hathim ar-Razi apparently didn't know him, a-Dhahabi and al-Khateeb al-Baghdadi regarded him as trustworthy while ibn Hajar al-'Asaqlani mentioned a-Daraqotni's qualification saying he is of trust (sadooq) but he makes mistakes, but a-Darqoqtni also said about him once he is matrook (See also in the database).
So finally we reached a rather known narrator, however again only al-Bukhari reported from him and none of the authors of the 4 Sunan, nor Muslim nor Malik.
The chains of our 3 sources
These are not all the paths with which this hadith was transmitted, but all of them start from one of Zakaria ibn Yahya's (abu as-Sukayn) students: Ibn 'Asakir referred to two narrator chains one of them via Abu Bakr a-Shafi'i, while the second was transmitted via the student al-Qadi abu 'Ubayd 'Ali ibn al-Hussayn ibn Harb ibn 'Isa أبو عبيد علي بن الحسين بن حرب بن عيسى and both al-Hakim and his student al-Bayhaqi reported via a fifth path via abu al-Bakhtari.
A qualification of the narrators
The chain part of ibn Qutaybah:
- Yazeed ibn 'Amr al-Ghanawi was only accepted as trustworthy by ibn Hebban in his a-Theqaat الثقات - where his qualification is rather lenient.
Note that some scholars among them al-Hakim and others accused ibn Qutaybah's trustworthiness as a hadith narrator and others in his 'aqidah. Imam a-Dhahabi quoted their claims in his Siyar 'Alaam an-Nubalaa', but also informed the reader that none of them came up with proof nor could any of this be concluded from his work.
Further note that the person who is reporting the book is one of ibn Qutaybah's sons most likely Ahmad (he is known to have memorized all his fathers books) who died 322 a.H..
All this shows that the version of ibn Qutaybah has issues in almost each level.
The chain part of abu Bakr a-Shafi'i:
Similarly to ibn Qutaybah he has only one intermediate between him and the chain as described above and here he names two people who had transmitted the hadith to him:
- abu Sheikh Muhammad ibn al-Hassan al-Asbahani died 286 a.H. and was accepted as trustworthy by al-Khateeb al-Baghdadi. he is one of the students of Zakariya ibn Yayha at-Taa'i and a teacher of abu Bakr a-Shafi'i (see here in the database).
- 'Abdullah ibn Muhammad by checking both the teachers of the author abu Bakr a-Shafi'i (database showed 9 potential matches) and the students of Zakaria ibn Yahya (database showed 4 potential matches) the best candidates for being the person we are looking for are either:
- 'Abdullah ibn Muhammad ibn Najiyah al-Barbari who died 301 a.H. and is trustworthy or
- 'Abdullah ibn Muhammad ibn 'Obayd ibn Sufyan ibn Qays best known as ibn abi a-Dunya (209 a.H. -282 a.H.) who is accepted as sadooq or
- 'Abdullah ibn Muhammad a-Duri who deid 302 a.H. and is also trustworthy.
This chain seems to have a more reliable narrators.
The chain part of imam at-Tabarani:
Imam at-Tabarani provided the name of three teachers from whom he reported this report:
- Muhammad ibn Musa ibn Hammad al-Barbari (213 a.H. died 294 a.H.) his hadith is accepted as hassan or saddoq however a-Daraqotni doesn't regard him as that strong (memorization).
- Ahmad ibn 'Amr al-Bazzar the author of al-Musnad (died 292 a.H.) he certainly is trustworthy however many scholars had some criticism: Al-Hakim said he was making mistakes in his isnaad, a-Daraqotni said almost the same, adding that he relied on his memorization a lot and a-Dhahabi referred to him as sadooq and well known, an-Nasai had also his reservation towards him.
- 'Abdaan ibn Ahmad al-Ahwazy (216 a.H. - 306 a.H.) he was accepted as trustworthy and a known qadi of his time and known to have memorized 100.000 hadith.
Again this looks like a part with reliable narrators.
Note that neither al-Bazzar nor ibn abi Khaythouma -whom was mentioned by ibn Hajar as a possible source- seem to have mentioned this hadith in their known books.