I don't think the issue will be whether to have an HTT or a fully-enclosed TT (they're generally not much better sealed than a decent HTT), but rather whether you can reasonably camp at all. I suspect that for most HTT and TT are pretty much the same in terms of allergies.
There are not a lot of common/typical allergens in wood smoke - however if your son has asthma the particulates that form the smoke COULD be a big problem. None of us can judge this.
Another issue with most (all?) campers is the use of glues in their construction that off-gas various compounds. Most people are okay with them, but some are affected by them to greater-or-lesser degree and we can't judge whether your son will be one of those people.
Also, if a camper isn't carefully cared for (especially during winter storage - close a bunch of warm air inside, then cool it down - the moisture condenses and over time can lead to the growth of invisible mold in seat cushions and the like) or if it has ever leaked, it can develop mold. Whether this is an issue or not, and/or whether or not those used trailers have this problem, we can't say. Definitely something to watch out for though.
Talk to your son's doctor about it, if s/he says it's okay, give it a shot using a rent-a-camper and see how it goes. If it goes well, but you're still not sure about the HTT, rent one, or ask to try out the one you're looking to buy, and give it a shot before buying.
We have a Jayco 23B we just bought this year. Before that we had a Jayco 10UD pup. Our 12 year old daughter has allergies to many types of pollen as well as having asthma. She takes asthma medicine, several allergy medicines and antihistamines as well as she has an inhaler. We have gone out camping during several different times of the year in the pup from early spring thru late fall. She has never had an attack from sleeping in the pup or the 23B. She really enjoys the time of camping and the family time and always looks forward to going out.
Bill
DW Trish
DD Sam (12)
DD Mykayla (2)
2008 Jayco 23B
I have serious allergies but love to be out doors. About a year ago I was told about many years ago and never thought it would work. Before this I would cut the grass come inside barely able to breath and even after a shower it would still be bad. Now I rarely even sneeze. Go buy a jar of LOCAL honey and give your son a spoon full daily. It was an instance fix for me. Give it a try, just make sure it is local.
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