Seriously, that's a tough situation. I don't know about in other states, but in CA, pets are personal property same as furniture (I don't think that's the right way, but that's the law). My suggestion is to try to make an agreement, and maybe say that if you get visiting or borrowing time, you'll share some of the vet and food cost, etc. If she's being difficult, I'd just drop it for now, and wait.
I had a similar situation with my ex-GF (with 2 dogs), and since one thing that we always agreed, is that the wellbeing of the dogs is #1, it worked fine. I'd keep them for a few weeks, and then she would, etc. This went on for 5 years, and then she got a new puppy, and she worried that the puppy would be too much for the older small dogs, so she asked me to take them permanently. All good, and she can see them anytime.
Charles