I wouldn't give an asthmatic child anything without discussing it with his doctor!
You really don't know--for sure---when his mother treated him and he could have a drug reaction. She may have given him meds just before you got him to keep you from complaining. His meds may not be working and need to be changed.
When my grandson was young we had to monitor him with his doctor on a regular basis and sometimes changed meds. A child can go into a full blown asthma attack in a matter of minutes! That is the most frightening thing I have ever experienced. (if he ever begins to strain to breathe and become suddenly lethargic, call the EMTs immediately!) An untreated sinus infection can send me to the hospital!
Your husband should have a letter, from the mother or the courts, stating that he is the legal father and has the right to seek medical treatment. Then as soon as you can, take him to visit his doctor and explain the situation. Doc will tell you what you should do.
And, C., I detect a lot of self rightous indignation in you. Yeah, I've done that too! You are not going to change his mother so don't waste your energy.
Enjoy your stepson! Children grow up way too fast. (Unless you have 3 under 3 years old. LOL!!)