Short answer: Du'a from the qur'an if regarded or intended as du'a doesn't fall under this case. And this is agreed upon among scholars.
A superficial analysis
First of all the chapter "The Prohibition Of Reciting The Quran While Bowing And Prostrating" in Sahih Muslim starts with ahadith showing that among the last things the prophet () said before he died was:
The Messenger of Allah (ﷺ) drew aside the curtain (of his apartment) and (he saw) people in rows (saying prayer) behind Aba Bakr. And he said: Nothing remains of the glad tidings of apostlehood, except good visions which a Muslim sees or someone is made to see for him. And see that I have been forbidden to recite the Qur'an in the state of bowing and prostration. So far as Ruk'u is concerned, extol in it the Great and Glorious Lord, and while prostrating yourselves be earnest in supplication, for it is fitting that your supplications should be answered. (Sahih Muslim 479a and 479b)
Even if in these two narrations the focus was more on the topic if the glad tidings of apostlehood the best recommendations for ruk'u and sujood are mentioned. Then it moved on with narrations telling us that the prophet() forbade the recitation in both:
The Messenger of Allah (ﷺ) forbade me to recite (the Qur'an) in a state of bowing and prostration. (Sahih Muslim 480a, 480b ,480d)
Then it goes on with a narration that seemingly implies that this was ordered to 'Ali():
The Messenger of Allah (ﷺ) forbade me from the recitation (of the Qur'an) in bowing and prostration and I do not say that he forbade you. (Sahih Muslim 480c)
Then it goes on with narrations that didn't mention sujood
This hadith is transmitted on the authority of 'Ali, but he made no mention of" while in prostration". (Sahih Muslim 480e -actually mistranslated I've reported this-, 480f)
Note that the hadith you've shared (480e) is narrated via a bunch of routes all of them starting by 'Ali() or ibn 'Abbas() from 'Ali(). In fact all the ahadith 480a-f are on the authority of 'Ali, while 481 is on the authority of ibn 'Abbas and shows a difference, between most of the earlier narrations.
The scholars are of the opinion that recitation of the qur'an is at least frowned upon during bowing الركوع and prostration السجود.
A deeper dive
Some scholars even tried to explain the wisdom behind this:
It was said that the best pillar of prayer is standing and the best of dhikr is reciting Qur’aan, so the best has been put with the best, and it is not allowed to put it with anything else, lest anyone think that it is equal to other kinds of dhikr. ‘Awn al-Ma’bood.
It was said that it is because the Qur’aan is the noblest of speech, because it is the Word of Allaah, and the position of bowing and prostration is one of humility and submission on the part of the worshipper, so it is more appropriate not to recite the Word of Allaah in these two positions. Majmoo’ al-Fataawa, 5/338
(Source islamqa #46997)
There's a difference of opinion whether this hadith must be understood as an order and therefore the recitation is prohibited in the prayer while bowing or prostrating and if done the prayer would be invalid. Or whether it is "only" frowned upon and therefore if a person does so the prayer is still valid. This later view is that of the majority and one of two statements of imam Ahmad.
However recitation is not equal to recitation: while standing in the prayer one must recite the qur'an this is in consensus of all scholars however whether reciting the al-Fatihah is fard is also discussed. But in other prayer parts of positions it doesn't seem convenient to recite the qur'an in general. However suplications from the qur'an are a clear exception.
If a person recited a du'a from the qur'an, like:
..."Our Lord, accept [this] from us. Indeed You are the Hearing, the Knowing." (2:127)
..."Our Lord, give us in this world [that which is] good and in the Hereafter [that which is] good and protect us from the punishment of the Fire." (2:201)
...My Lord, make me an establisher of prayer, and [many] from my descendants. Our Lord, and accept my supplication. (14:40)
our Lord, forgive me and my parents and the believers the Day the account is established." (14:41)
or a variation turning plural into singular and vice versa, during his sujuood (as performing du'a during sujud is recommended -see for example in Sahih Muslim-) intending it as a du'a (suplication) it is fine and acceptable. As the intention is the key for every deed as stated in the known hadith.
Al-Zarkashi said: It is makrooh when it is intended as recitation of Qur’aan, but if it is intended as du’aa’ and praise, then it is like someone who did Qunoot by reciting a verse. End quote.
Imam an-Nawawi further commented on 480c (See -here- in Arabic)
In the following I'm translating from Arabic language, as these translation are of my own take them carefully!
قوله : ( نهاني ولا أقول نهاكم ) ليس معناه أن النهي مختص به وإنما معناه أن اللفظ الذي سمعته بصيغة الخطاب لي فأنا أنقله كما سمعته وإن كان الحكم يتناول الناس كلهم
His saying: (He forbade me,... and I do not say he forbade you) does not mean that the prohibition is specific to him. Rather, it means that the word that I heard in the form of the speech is mine, so I transmit it as I heard it, even if the ruling covers all people
Therefore I used the term "seemingly" above.