nisht: I’m disappointed in myself also. I have to remember, but someone quoted that the chicken should Davka go to Tzedaka. I apologize for thinking of the wrong one.
The Minhag isn’t that long-standing, by the way. It started well after Rashi’s time (see the Rashi I quoted as the second comment in this thread). The Mechaber calls it a Minhag Shel Shtus (before later printers edited it to not be mean to the Ashkenazim). It had major questions on it to begin with (which Pashtus is we answered because Poskim for the most part defend it, not scream at it) and the fact that a huge factor of Tza’ar Ba’alei Chayim is now included should really matter to us.