Hi OP,
I think oftentimes our fixations on things, particularly a partner's pet that they had before us, are informed by deeper fixations/insecurities/worries that we have. They almost become the 'step kid' in the relationship.
Did you have the pet in a previous relationship? Does he associate the cat as being a previous partner's, and therefore not his? Can you think of other issues that he is bringing up before he launches into one of these tirades about the cat?
Sometimes when people don't feel heard, they can then shift focus to things that they are able to control, and unfortunately pets are often caught in the crossfire of this misdirected angst. It's possible that your DH doesn't even realise that he's doing this.
In no way does it excuse his behaviour, but I think trying to make sense of it will make the situation less maddening for you, and may help you to have a meaningful discussion with him that has more of a long-term fix, rather than a short-term fix if you've only dealt with the surface issue i.e. his annoyance.
Next time an episode of this is triggered, ask him about what his other grievances are at home/in the relationship, and try to work on them. If you're finding that too tough, consider couples' therapy.
Hope this is helpful, OP. Hang in there 💐