In terms of the test . . . nothing wrong with doing it to ease your mind or if your DS needs it for his ongoing care. However, in all honesty, it's unlikely to have much if any bearing on his military application (SA or ROTC).
What will PROBABLY happen is that next year your DS will fill out his form and take his test and . . . get a remedial. DODMERB will then ask for additional medical records and/or request certain tests -- whatever test they consider appropriate/necessary. Based on that info, DODMERB will either qualify or DQ him. If he's DQ'd, it will be up to the individual SA (or ROTC program) to waive that DQ.
As a general rule, if a candidate has symptoms of asthma or is diagnosed with asthma after age 13, it is likely a DQ and tough to get a waiver. Using an inhaler isn't as critical as the fact that an MD believes one might be necessary. That said, people here will cite exceptions to the "general rule" -- and you won't know your answer until you start the process. As the Dep. Dir. of DODMERB (who used to frequent these boards) so often said: every situation is unique.
Unfortunately, there is nothing that you or your DS can do at this point to get a better feel of what will happen next year. I recommend your DS complete his application as quickly as possible next year so that he gets "invited" for his medical earlier. That way, you should have an idea early on regarding his chances, medically.
And, as with every candidate, have a plan B that -- in his case -- doesn't include the military.