I am so sorry that he is acting childish over everything including the dog. He is angry and he wants to control everything and like all of the posters have said...that's his anger to own. I know it hurts to know that he's pulling out all of the stops to act like a bratty child, but do not allow him to see you sweat. Your lawyer is on the ball here and he knows what to do and will not allow your h to walk all over you.
As a poster pointed out...you were married and anything that was purchased/earned during the marriage is considered marital assets. These MLCers don't get it and are very stubborn about what they think that they are entitled to. If he wants the dog, so be it...but he's not going to want to take care of the dog for very long because it involves responsibility. Here's my take on it, the more you try to point out that it's your time w/the dog, the more he's going to want to take that dog away from you. He's acting like the bully and/or kid who sees another child w/something he wants to play with and yet the kid playing w/the toy won't give it to him, so he has to become the bully to take it away from you. Once he gets the toy, he'll grow tired of it and then leave it where it falls. It's called being a bully and attempting to control the situation and hurt you in the process. Again, don't let him see you sweat.
Choose your battles and if it means him taking the dog...let him. You have bigger battles to fight right now.
I do hope that you can enjoy your Thanksgiving.
Sit quietly, the answers will reveal themselves when you least expect them to. The past is gone, the present is a gift and you need to focus on today, allow the future to reveal itself when it is ready.
