Ugh I dated a girl very much like this (not quite as extreme though) and while it wasn't the main reason I left her it was absolutely a huge part of it, and like you I didn't bring it up because she loved that dog more than anything & criticizing her for it would've gutted her.
I think being "Alpha", meaning competing, with the dog was part of the IA's point. Having to compete with a dog would be pretty humiliating. The image the IA's last sentence created was pretty funny, though.