If you've got asthma I'll buy everyone on Jet Blast a drink.
(Note: that was a joke and does not constitute a contract).
Whilst the AME's preliminary role is to protect the public from sudden incapacitation , they do actually like to pass pilots fit to fly.
From what you've posted on here, there seems to be no definitive diagnosis of "asthma" in your case. The clinical diagnosis is not always clear cut, and historically the "asthma" word was sometimes banded around a little freely. Nowadays, I prefer to use terms such as "viral wheeze" etc. They will be more interested in your history, (episodes per year, hospital admissions, severity etc).
I suspect that the authorities have to cover their arses, which is fair enough.....it'll probably cost you a few hundred quid, but, I suspect, money well spent in the long term.
Let us know how you get on. Good luck with your career. :-)