Note that this incident happened after fath Mekka.
There are several ahadith quoting the words:
Fatima is a part of me. He in fact tortures me who tortures her. (sahih Muslim)
إِنَّمَا فَاطِمَةُ بَضْعَةٌ مِنِّي يُؤْذِينِي مَا آذَاهَا
One which might get close to the one you posted is:
I heard Allah's Messenger (ﷺ) who was on the pulpit, saying, "Banu Hisham bin Al-Mughira have requested me to allow them to marry their daughter to 'Ali bin Abu Talib, but I don't give permission, and will not give permission unless 'Ali bin Abi Talib divorces my daughter in order to marry their daughter, because Fatima is a part of my body, and I hate what she hates to see, and what hurts her, hurts me." (Sahih al-Bukhari)
an other quoted the reason for the refusal:
'Ali b. Husain reported that Miswar b. Makhramah informed him that 'Ali b. Abi Talib sent the proposal of marriage to the daughter of Abu Jahl as he had Fatima, the daughter of Allah's Messenger (ﷺ), (as his wife). When Fatima heard about it, she came to Allah's Apostle (ﷺ) and said:
The people say that you never feel angry on account of your daughters and now 'Ali is going to marry the daughter of Abu Jahl. Makhramah said: Thereupon Allah's Messenger (ﷺ) rose up and I heard him reciting Tashahhud and say: Now to the point. I gave a daughter of mine (Zainab) to Abu'l-'As b. Rabi, and he spoke to me and spoke the truth. Verily Fatima, the daughter of Muhammad, is a part of me and I do not approve that she may be put to any trial and by Allah, the daughter of Allah's Messenger cannot be combined with the daughter of God's enemy (as the co-wives) of one person. Thereupon 'Ali gave up (the idea of his intended) marriage.
(sahih al-Bukhari, the one you found from sahih al-Bukhari and in sunan ibn Majah)
so the main reason is not the Prophet's refusal for polygyny, but his refusal to see the daughter of the enemy of Allah (abu Jahl) and the daughter of the servant of Allah (Fatima bint Muhammad) being co-wives of the same husband!
See also fatwa islamqa #162287
Scholars explained this hadith based on the verse (33:53):
... And it is not [conceivable or lawful] for you to harm the Messenger of Allah or to marry his wives after him, ever. Indeed, that would be in the sight of Allah an enormity.
so the major issue according this view is that it is not lawful to harm the Prophet (). and as he made clear that hurting or harming Fatima() is equal to harming him, it was not lawful for 'Ali() to marry abu Jahl's daughter.
Ibn al-Jawzy said it is possible that the Prophet () made it a condition for marriage that 'Ali shouldn't marry an other wife, while married to Fatima().
For more details see the Arabic fatwa islamweb #163241.
Also note that in other versions of the hadith the statement:
I do not make a legal thing illegal, nor do I make an illegal thing legal
(sahih al-Bukari, sahih Muslim and sunan abi Dawod)
appears, which is a clear statement against the claim, that the Prophet () discouraged polygyny, but makes clear that we have a special case here!
Note that in 'ilal a-sharai' (meaning "the cause of situations") Volume 1 pages 185-186 Sheikh Saduq quotes a similar quote affirming the story about the proposal for Juwayriya the daughter of abu Jahl does actually appears in Shi'a sources too:
فقال رسول الله صلى الله عليه وآله يا علي أما علمت أن فاطمة بضعة منى وأنا منها فمن آذاها فقد آذاني من آذاني فقد آذى الله ومن آذاها بعد موتى كان كمن آذاها في حياتي ومن آذاها في حياتي كانكمن آذاها بعد موتى
(My own translation take it with the necessary care)
And the Messenger of Allah () said: Oh 'Ali, don't you know that Fatima is a part of me and I'm part of her. He who in fact hurts her hurts me, and who hurts me hurts Allah. And he who hurts her after my death is as one who hurted her during my lifetime and he whom hurts her during my lifetime is as one whom hurts her after my death