I know how you feel, to a certain extent.
I used to own horses, they're my dream animal and still to this day are but a fair few people in my family are allergic to them and I wanted to have a lot of them watch me go riding and competing. May seem a bit ridiculous but you'd be surprised how much hair can float around inside an arena, even if it's an open arena. The sheltered seating area can catch a lot of that hair. I can't own them anymore due to finance issues (and stabling them is expensive each week by itself) so I had to sell them up after one of them passed away in 2016 and wait until I get a bit older and get myself stable to own them on my own property. (I hope to one day achieve with stables and a paddock)
As for pets, there's a few I can recommend but some may be too 'exotic' for you.
- Parrot (budgies, macaws etc.)
- Ferret
- Rats; I know you stated your son is scared of them. But maybe showing this video may help. Bean was such a lovely rat. Before I saw this video I didn't acquire to owning rats... now I can't stop thinking about getting two or more (they are social animals and require a pair or group). But it was a little too late as I had just stepped back into the world of owning hamsters. I don't have the funds for rats on top of the pets I have.
RIP Bean - Tribute to the Most Amazing Little Rat �� - YouTube
- Chipmunk
I would suggest a Sugar Glider, I want one of those as my pets too after studying during my degree. These are really exotic though and although they don't require a license they can be pretty hard to care for, especially for a child.