Another vote for this being essential to discuss before he moved in, never mind before you had a child. Now the horse has bolted, so to speak, IF you see the relationship lasting, then imho you need to suck it up. Of course he has a responsibility to his cat.
My now-DH moved in with me and two dds a year ago. We discussed his much adored (and thankfully elderly) cat beforehand - I emphatically didn't want a cat, but accepted that they came as a pair. This was on the firm understanding that once this one had gone, there wouldn't be more, which he happily agreed to and which is 100% being stuck to. We have a 9 year old dog who he adores and who utterly adores him, by the way.
I had no antipathy towards the cat herself and was careful to look after her and keep my end of the bargain (though he did the vast majority of Cat Duties) - but I absolutely hated having a cat, especially an elderly one. It really started to upset me. Ended up with a litter tray in our bedroom - as well as other places - for the entire duration (it was the only place she could be shut in for the first few weeks: doubly important given the dog). I loathed the whole thing, esp as she would stand in the tray and pee outside of it, so we had acres of pee pads everywhere and despite endless changing, the smell really got to me. Dds and I had to constantly remind ourselves it wasn't forever, and were very careful not to let any anti-cat speech out in front of dh but it was horrible tbh.
A couple of months ago the cat was diagnosed with kidney problems which explained what was by then endless peeing everywhere. After responding beautifully to the treatment for a few weeks, she suddenly went downhill very quickly and had to be PTS. Dh was devastated, bless him. I was upset for him but very relieved. But I love him and fully respected his need to look after her and honour his commitment to her.
It all depends on your relationship. I was lucky it wasn't for that long in the end, but I love him partly for how kind and caring he is, and wouldn't have dreamed of making him choose as he'd had her for 14 years.