I don't have a suggestion on what he might be allergic to but I do have some other comments to your post.
It is true that if you did a follow-on allergy test and the child had recently taken antihistamines, the results won't be accurate. I think you need to be antihistamine free for something like 72 hours beforehand.
When washing the laundry try a double rinse and no fabric softener. For the sheets wash in hot and definitely get allergen protectors for the pillow and mattress.
My cat allergy did not develop until I was around 10 or 11. It started with long hair and then expanded to all cats. Now sometimes I can just walk into a house and know they have a cat even if I don't see evidence of one. The dander can get into everything that you took with you and be in the carpet for months if the people living there beforehand had pets.
A positive allergy test result does not indicate what type of reaction the person has. One could have a strong/high test result but that does not indicate how bad the person's reaction will be. For example, my cat allergy test does not give the strongest result, but it is by far my worst allergy. I test positive for allergic to cats and dogs. With cats I just have to be in a house that has them to feel the effects. For a dog, I have to physically touch the dog (or dog hair) and then touch my face.
I'm also reluctant to give my kids ongoing medications especially if it's unclear if it's what they really need. However, I think it's worth trying the medicine to see if it helps. I don't know if a month is necessary, but at least a couple of weeks.
FYI, I have issues with cinnamon as an allergy/asthma trigger and it does not show up as a positive reaction on an allergy test.