My roommate wants a cat and I'm not going to allow it to happen.
Why do you have a problem with them getting a cat?
I love cats, but I'm allergic.
That's fair to you. You don't want to be sneezing and itching all the time.
Yeah, I think being allergic is a valid reason to say no.
It's unfortunate that you're allergic, but have a reaction to a cat is a miserable experience that I genuinely understand. Let your roommate know you wish that they could get one, as you love cats too, but it just can't be due to your own health.