I have asthma and am allergic to Timothy Hay. I knew this BEFORE I got Pepper, and part of my prebunny planning was figuring out how to manage things.
Some things that I do that help:
1) See a pulmonologist about your asthma, and get allergy tested. You need to know what your triggers are and how to manage your condition before jumping to comclusions as to what is setting off your symptoms. I’m assuming your “puffer” is a rescue inhaler like albuterol? The goal of most asthma treatment regimens now is to minimize the need to use those by preventing symptoms before they start. Antihithamines like OTC claritin or zyrtec, anti-leukotrienes like Singulair, and inhaled anti-inflammatories like Advair, FloVent, or Intal can head off your symptoms BEFORE you get an attack.
2) Vacuum regularly. I have a Roomba and have gone through 3 sets of brushes, but it’s a godsend
3) Use a hay rack above the litterbox – it keeps the hay mess contained and the bits mostly just fall in the box. I only have to refill it every week or two. Also change the litter/cage materials at least once a week.
4) Get and use a HEPA filter in your bedroom.
I have been asthmatic for more than 25 years now and treatments have changed a LOT over that time.