This is a very, very typical response when the WW feels pressure lifted. The minute her H agrees to her wishes, she starts saying this type of mush. It means absolutely nothing!
I suggest you let her have the dog. It's not a child, and co-parenting pets just cause opportunities for more hurt.
Brace yourself for her to act like your BFF. It's just a WW's reaction, nothing more. She loves the dog more than you, so don't be tied to her with a pet.
It's difficult, but this is part of it. She is going to take this time & space, whether you agree and accept it or not. I suggest you do nothing to slow her down. I can't remember at the moment if she works, but apparently, she expects financial support from you. Do what the law states, and don't give more than that.....trying to soften her. She'll take as much as she can get. Be smart.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!
