I think you can see a lot about a person in how they treat animals. I don't care if they "like" them or not- a respectful, kind hearted person would not treat an animal terribly even if they didn't like them.
I understand that you say it's your husband's culture to not treat animals with respect, but is it part of his culture to force someone he loves to make a touch decision that could ultimately break their heart? I really can't wrap my brain around that one. My husband may not like some of the things I do but he ALWAYS puts my happiness first. Honestly, your FI can't handle your cat but he's willing to make you sad over it? That's something I would think about.
Other girls have mentioned the control aspect too. If I were you, I would worry what other compromises you would have to make to make him happy.