Yes, when I was younger there was no effective treatment so I hated 'PE' and always had 'a note' or hung around the edge of the pitch trying to avoid the ball.
Nowadays the medication is so good I'm sure your Asthma will be brought under control, try to use that to get fit then you can experiment with backing off the medication.
My routine to get fit was to use the steroid sprays to the letter of the prescription then two puffs of ventolin one minute apart ten minutes before my run.
Keep at it and good luck.