Thank you for the deep dive into the ugly place. I did the same when I read this book several months ago: https://tuckerlieberman.medium.com/trans-kids-undermined-irreversible-damage-regnery-6b425e6c3fe7

As for what to make of the comparison of being transgender to the claim that one is an animal (a chicken, a cat, etc.), I've argued elsewhere that it is a version of the "slippery slope" fallacy. Not too long ago, people tried to seriously argue that same-sex marriage would lead to people marrying their dogs; it did not. People say that gender transition will lead to people identifying as animals; well, gender transition is not new, and people are not identifying as animals.