I have had mild asthma since I was 9 years old. It was at it's worst in my late teens, when I had to use a nebuliser (machine with mask) several times a day but that only lasted a year or so I think. If I take my preventer (I used to be on becotide but now I'm on flixotide) when I'm supposed to I almost never need to use my ventolin but I never take it! My husband gets so frustrated with me but I'm just really forgetful and tbh it doesn't bother me to take ventolin every now and then, I just always have one in my handbag and another by my bed.
So I wouldn't worry about it at all - it's a minor inconvenience but won't do your LO any harm and honestly I have never felt handicapped by my asthma at all. There are MUCH worse things you could have! Hopefully he will grow out of it (which I think is more likely in children who get it really young as opposed to my situation) but if not it's honestly no big deal.
C

PS - I am confident that BFing has nothing to do with it