I recently got myself a cat and found myself with the asthma dilemma myself.
I'm allergic to most cats and stay well clear. My asthma is solely triggered by animals or sport, so I have avoided both all my life and never needed an inhaler. For the one-in-5 year situation something brings it on, I borrow a family member's.
Then last year my parents got a bengal cat. I was sure i'd never be able to visit but to my surprise I was 100% fine. I lived in with house with him for a month while they went on a cruise, I totally fell in love with the cat. All fine. So finally i thought I'd found my breed!
Cue - my own bengal last November. Within 2 weeks I'm coughing and short of breath and absolutely beside myself with upset on what to do.
I've got inhalers from the docs, I'm a bit in two minds about having to take medication for the sake of the cat but I'm giving it a go. In a few more weeks I'll allow her outdoors and I'm REALLY hoping being able to have windows and doors open will improve the situation.
Otherwise me and my parents may have to do a cat swap!!
OP you have my sympathy.
Cats find rehoming stressful but only temporarily so. It will not cause lasting damage to your cat to rehome if you need to, but I'd be tempted to give yourself longer before giving up on the idea. It will be temporarily sad for your kid too, but also a lesson in funderstanding everyone's needs.
Also - if you haven't already, keep the cat out of the bedroom, that honestly makes a big difference!