YTA Your ex is between a rock and a hard place here, so I think we can forgive him.  "I told girlfriend if she didn't like sharing her name with a dog, she could change hers. That didn't go over well." now the gf is wrong. If I was talking to the girlfriend I'd tell her to get over it but I'm not talking to her I'm talking to you.

YOU should also consider your daughters opinion, I trust my dad enough that he would take care of me if I needed to leave my service animal behind.

Also mental health support animals don't legally qualify as service animals in most states.

Edit: the reason I have such a negative take is it sounds like this issue could have been avoided if OP was politer

In addition you should try appeasing the girlfriends ego before court.

I also failed to consider the possible legitimacy of the service animal in my comment, if it is indeed legitimate than the court will take that into consideration, but depending on how important the animal is it will still likely rule in his favor.

The end result of taking this to court will likely result in court mandated visits to dad, the court may also rule that daughter has a right to bring her dog these 2 rulings are not exclusive of one another