Short of having an asthmatic attack at altitude, it's probably not going to be an issue operationally. Getting past the paperwork, however, could very well be. I don't know what Australia provides with respect to Asthma, but in the US it's considered childhood asthma if it hasn't been repeated since the age of 12. If the paperwork reflects a statement that asthma hasn't been experienced since the age of 12, that may well do the trick. If you tell the examining physician you have asthma...then you're opening up a can of worms.
I'm an asthmatic. I grew up at density altitudes in the summer of well over 10,000'. I have more problems breathing at sea level than at higher elevations, and I've found that I tend to handle an oxygen deprived environment better than most; I spent much of my early life in that condition. I worked as a firefighter in the smoke for years, and I have regularly been above FL180 doing parachute jumps without any difficulty. Don't assume that being asthmatic automatically pushes you out of the running, or disables you at altitude. I've experienced on decompression at altitude without any ill effects. A much bigger concern would be a sinus block at altitude, which has nothing to do with asthma, and is something to which everyone is subject.
Truth be told, if one is truly a severe chronic asthmatic, an oxygen deprived environment is familiar territory. I know.