As much as I love animals, that's a ridiculous excuse. Tell her that you'll deliver the cat to her when she moves, or call her brother and arrange for yourself to bring the cat to him, or see if there's a neighbor or vet who can help. People who are as bad off as the two of you are don't stay together even for the sake of their own children, and a cat who may or may not be sick and may or may not like their new home and who she IMAGINES might not handle a 4 hour drive (so stop several times if it's so distressing to the cat), good grief.

She's going to try every trick in the book. Talk to Kent. His ex is the QUEEN of trying every trick in the book... or ... well, I think HK, who's ex is trying every trick in the book and is succeeding from time to time.