Looks as if everything may need to go through the lawyers b/c contact with her is hopeless. She will never stop taking every shot she can get at you. Now that you have a girlfriend, she will really get ugly b/c it's eating her up.
Don't say anything else about the cats.
You may need to appoint someone to be the go-between in the needed contacts regarding kids. Don't give opportunity to get drawn in. I know couples who got D and never contacted each other over kids information again. At least not by phone or email. She doesn't want a D b/c she's had all this power over you by using the kids as leverage. Once D, she will have to let you see them as the court dictates.
If you could learn to laugh at her and say something like, "If you say so", and she saw she couldn't push your buttons, she might stop......but I don't know. Heck, she pushes my buttons!
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!