I'm not a doctor but I was diagnosed with asthma 25 years ago.
Sounds to me like you may need a more effective steroid preventer?
I suffered with asthma all through school and my teens because I wasn't told to take my preventer correctly. Like muddy@rseguy, I've found that taking my preventer (Becotide) regularly made an astonishing difference. I used to use my Ventolin all the time, because I wasn't taking my preventer regularly. Since I've been taking my preventer every night I've had to use my Ventolin only 7 or 8 times in 5 years.
Have you been allergy tested? If you are allergic to something you are exposed to regularly your histamine levels could be elevated, making your asthma worse. When I had blood allergy tests (and did an exclusion diet) I discovered I was allergic to cows milk, amongst other more obvious things like cats and dogs. I since found that a lot of the medications I was taking (the old style powdered Ventolin and Becotide, all antihistamine and most other drugs in tablet form…) contained cows milk so they were prolonging the agony!
It may be worth having another chat with your doctor. Does your doctor even know you're a cyclist, who's thinking of racing?