We had a similar dilemma with our young child. If I could have a "do over" I would take the child to an MD specializing in Integrative medicine. I know you had your child allergy tested, but if it was like what our allergist did, it was all IgE testing. I didn't even know there were other types of possible reactions. Because of that, it really derailed my child's life when there were things we could have changed simply way back when. (you might be interested in this article:
It's Not Mental: Allergies, Asthma, and “Mental” Illness and the book you see there is relevant.) Anyway, she is now on some allergy meds, but off the asthma meds - down to 1 plus 1 prn rather than a bagful of different things. Still a bagful, actually, but in the form of fish oil and other supplements. Used to get sick so often and now not so much.