There is very little protein in peanut oil. So little, in fact, that it's considered to not be allergenic. There are people on this board (myself included) who do react to the minute amount of protein in peanut oil. There are also people on this board (again, including myself) who have airborne reactions. However, I think the possibility of having an airborne reaction to the minute amount of protein in food cooked in peanut oil would be extremely slim.
Sometimes asthma is just asthma. I know your DS doesn't usually have problems with environmental allergies, but pollen is especially bad this season in many places plus we're moving into cold and flu season. I'd be more inclined to think him needing the inhaler was due to that.
Hoping your DS is better today and also hoping there's a simple explanation for his asthma flare!
ETA: Just seeing maeve's post about the peanut dust - yes, could definitely be that! I didn't realize Five Guys had the peanut shells.