The gamble is:
Tell the truth, and risk missing out on one career path among many.
Lie, and risk being found out, in which case you can kiss goodbye to the career altogether.
Sorry to be so blunt, but lying isn't something you should be considering; it doesn't tend to go down too well if you contradict your medical records without a very solid reason.
As Warty99 says, though, maybe you should consider talking to your GP before your medical about whether he could record the asthma as a misdiagnosis? Lots of people were diagnosed with "mild" asthma in the 80s and 90s, when it seemed almost to be in fashion; not many of them will ever have had an asthma attack or anything to prove that the diagnosis was correct. Since your desired career might hang on this, maybe a polite chat with your GP is the best way to go.
Whatever happens, best of luck for your application.
IANAD.