Sadly as you know some people do develop bad allergies to hedgehogs and need to rehome them due to the issues.
I will post a quote from another recent post from Nancy:
Sadly it sounds like you are allergic, especially to his poop or urine. As I have mentioned before to other people, I was allergic to my Smokey's urine. Lucky for me he was totally pee trained and as long as I replace the paper towel in his litter box daily, I was fine.
Do you have allergies to anything, foods, weeds, pollen, perfume, or anything at all? I know of a couple of people who were allergic to their hedgehog when the hedgehog was eating certain foods. For the one person, she was allergic to wheat and grains so changing hedgies food and she was fine.
Can you figure out if it is urine or poop that bothers you? Babies tend to have much stronger smelling urine and poop than adults and as he becomes an adult your symptoms may lessen but respiratory allergies, especially asthma is not something you want to wait too long on.
Is there someone in your home that could take him out and let him get his potty business over with before you handle him?
As a very long shot, there are products, usually made for ferrets that you can put in their water that will help control the odor from feces and urine. I don't know exactly what it does but it has helped some ferret people who were allergic to their ferrets business. It might be worth a try. It's called BiOdor or something like that.
Obviously the stool part does not seem to apply it might be worth discussing with a physician to see if anything can be done.