Gees, you could have been talking about me there!
I've got asthmatic tendencies and have just gotten over a mild does of bronchitis following a cold.
I didn't worry about seeing a doctor this time round as it went after a week. If the coughing lasts more than a week I cart myself off to the doctors and tell them the story. The smart ones check the peak flow and bung me on the Ventolin or if it's really not responding, a course of steroids to calm every thing down.
If you don't tell your doctor the full story then they'll just treat it as "bronchitis" and not manage the whole condition.
The doctor that made the initial diagnosis way back in 94 is also a DAME (aviation doctor). He said my version of asthmatic tendencies is not a limiting item on the medical so long as I take care of myself. Meaning, keep in reasonable condition, don't smoke etc.(Australian CASA)
The one question he asked that no other doctor had asked "have you ever had a bought of coughing that lasted more than a month for no apparent reason?" (ans, yes!)
He's Irish born and he recommended a session in the pub. His thoughts on modern cough medicines was hilarious. He reckoned that they are rubbish because the manufacturers took the alcohol out. He felt I needed a good night's sleep and a few beers would do that just fine. That was my prescription (along with the steroids).
I have been subscribing to that ever since
Good luck, see if you can find an enlightened doctor or tell the one you see all the time the whole history. They may be able to sort it from there. (I hold a class one in the UK too)
Your medical should be safe so long as you can pass the peak flow test and the coughing itself doesn't debilitate. I would see a doctor that isn't your aviation one and discuss a strategy. Asthma (however mild) is best managed rather than endured.