I can understand the "it's this or nothing". And like you said, it's not ideal, but it'll just mean you have different expectations and management needs. It's not THAT bad.
Like everyone said.... just watch for excess cool moist problems. I think it'll work out fine. And considering you want the honey for your son for allergies, you may want to do things specific for that. Like harvest all the honey you can, and feed sugar water all winter. You know best how much you need.
I think they CAN live, and live well, just might need certain care. Possibly. Or we might all find they thrive beautifully.
And yes, I think you can get more than 5 pounds honey a year.