I'm a moderately severe asthmatic. Ventolin, Clenil 250 (steroid preventative 2 doses 4 times a day), 4 sets of different asthma tablets plus another inhaler if it gets out of control, and steroids each and every time I get ill. I also can't have aspirin, ibrupofen (NSAIDs) or any dairy products. Many over the counter medicines are also out.
I still cycle, I still swim, I still hike usually in the winter.
As a result of my asthma medication, I have adrenal gland issues, I don't produce enough cortisol - goggle it and you will work out the implications - a very good reason not to double up on any of your medication (I didn't, it is just how bad my asthma has become - I have been resusitated twice so far, so don't under estimate your asthma).
I take my normal medication when out cycling, I don't double up. There are major long term side effects of preventers (though in my case I would be dead without them so there is not much point in worrying too much about the side effects!)
If you find that you are needing too much ventolin whilst cycling, speak to your GP. Ventolin can cause an increased heart rate, and weaker heart beats, so it is a balancing act. If you are taking significantly more when cycling, then speak with GP.
The bottom line, is that the fitter you are, the less it is a problem and
it should not stop you doing taking any exercise.
If you are struggling with your asthma whilst riding, ease off the pace and let the symptoms ease. Over a couple of weeks this should ease off completely, if it does not - GP visit.
It is harder in the morning at first wrt asthma and tight chestedness/wheezing and much easier by the end of the day in my experience. The one time I discussed this with my GP when I was much younger, I was basically told that was normal.
It is harder when it is colder and covering your mouth and nose does help. So does being fit.
I also find it harder, much harder when the trees are in flower or a thunderstorm is due, so learn to watch your warning signs...
I personnally would also recommend a
medical alert bracelet - especially if you are cycling alone like a commute. If you are in an accident, or having major issues with your asthma, you are not going to be able to talk. Pointing to a bracelet & turning it over if needed so people can see will quickly get you the help you need; the card that comes with it can details your medication/GP (if you want to) can also be very helpful for medical staff treating you.