OP I don't have DCs yet but I have some experience of 'blended' families and pets since my DM got a new DP a few years back.
We are all grown-up, unlike your DD, but one recent issue is that we all feel the DP is not nice to the family dog. For example, the DP will push the dog away if she comes up for a cuddle and call her smelly etc.
I find this annoying behaviour even though I'm an adult. I think if I were your DD, I'd be thinking,
'This is my home. This man just moved in here, laid down all these new rules that make me feel like a baby, and starts being nasty to my pet, who has been here way longer than him and whom I like better . He can piss off!'
Okay, a childish reaction - but she is 13 fgs, not 16 or 18.
I don't think calling the cat 'Sooks' is an issue but calling it fat is. It feels like a nasty dig.