Whilst I am not a doctor, I am a mild asthma sufferer.
As you do not need oral steroids to control your asthma, you should be fine for a Class 1. My AME knows I take the occasional puff on the blue inhaler, and by occasionally, I mean about once or twice a year. My asthma is so quiet it is said to be dormant now.
The only time the docs get a bit interested in your asthma is if it is not well controlled or more importantly, you go onto oral steroids to control it.
Be prepared for them to investigate it (eg a possible run around the building) but it sounds like you'll be ok.
Good luck.