I'd suggest a good translation to "butt-hurt" would be magoadinho/a, which is the diminutive form of magoado/a.
Portuguese definition of magoado:
adj. Que acabou por se magoar; que sofre ou expressa mágoa; em que há pesar; pesaroso.
Que possui nódoa; que sofreu uma contusão; contundido.
Que expressa ou sente dor (física).
Magoado/a, therefore, would translate to "(adj.) hurt" in English.
It is my understanding that "butt-hurt" has a derogatory tone to it, whilst also being a colloquialism.
Moreover, the diminutive form of words doesn't necessarily correlate to the "small form of the word", neither in English nor in Portuguese.
For instance: camisinha is not a small shirt; pegadinha is not (always) a small footstep; and both menininha and "girly" may be derogatory ways of saying that someone is too feminine.
Thus, magoadinho/a adds a derogatory, even sarcastic tone to being magoado/a, much like the transformation the prefix "butt-" does to "hurt" in the expression "butt-hurt".