If you don't mind my asking? which pump?
Depending on the severity of the asthma, a corticosteroid like flovent can help prevent attacks and ease breathing. Then a pump like ventolin can be used as needed for "attacks".
Again, I do not know if this is suitable for a child of 2, but I do know of adults whose asthma is well controlled using this regimen.
If your son sleeps alone in that second bedroom, if possible, for now at least, imo, it would be wise to keep the cat out of that room 100% of the time.