Hi Current
I am a physician AND an asthmatic. To make matters worse, my asthma is cold and exercise induced. I did however climb the mountaineers route in mid-April without much difficulty.
I recommend
1. talk with your physician about optimizing your pulmonary status prior to going. Even if you are not having symptoms, you may want to start taking your inhaler regularly before going. I did the first time but not the second because I was in much better shape overall.
2. Absolutely carry your inhaler (the albuterol-the quick acting one) with you at all times on the mountain. Use it at the first sign of tightness or wheezing.
3. Be as cardiovascularly fit as you can be prior to going. This will help a lot.
4. I took prednisone with me just in case. This oral steroid doesn't work as fast as i.v. but would definitely help 'calm things down' if you had more than a minor asthma flare- ask your doctor about carrying some with you. I was also able to take i.v. steroids but don't recommend this unless you are good at injecting stuff into veins (I won't ask

I didn't take any of the stuff but like to have it for myself or others who may get into trouble.
As far as the altitude and not the exertion, I asked the experts at a wilderness medical convention and they thought that overall, asthmatics did better with 'thinner' air- less resistance to airflow. Again, this doesn't consider the exertion that it takes to get there.
Good luck and with proper preparation there is no reason you can't do this.
-Cheryl