I'm also very allergic to cats (and horses! Also occasionally housedust). I have an inhaler which I only ever need to use on the occasions I'm exposed to cats etc. I don't really consider myself asthmatic, my GP and I describe it as allergy related asthma. Exactly the symptoms you describe. I really struggle to sleep in a house with cats, and need antihistamines and an inhaler if I visit someone with a cat. I think I barely use it more than once or twice a month, but I always pack it in my overnight bag along with anti-histamines just in case. I think of it like having paracetemol in the house - no biggie.
I'd recommend you visit your GP to discuss whether an inhaler would help you. An inhaler doesn't make you an asthmatic, and you don't get a dependancy or anything worrying like that. They just clear the wheeziness, (which can be quite frightening) almost instantly.
I don't think your allergy can "turn anaphylactic". It just sucks, and can certainly be made easier to live with!
HTH