If I'm being separated for unsatisfactory performance (failing two PT tests), does being diagnosed with asthma cancel the separation process?

It starts and ends with your unit leadership...So go see your 1SG (after requesting an audience though your platoon sergeant)
Start with, 1SG I WANT TO STAY IN....Im not looking to get out...(if that's true). I failed the APFT and I dont like that.(if thats true) ..Doc says I have treatable asthma (if you do), let me prove I can do this.... Hold the separation for 6 months, let the treatments kick in so i can do the work to make this right.

Unless you have already burned that bridge of good will, I'd bet the 1SG will take it to the commander and tell him to shelve the packet for a bit, lets see what the kid can do..or words to that effect.
The reg says the commander must "initiate " seperation,, that has been done.. the reg does not say you MUST be separated for APFT failure...that is something you will EARN..as will not being seperated be earned.

Have you ever been treated for asthma? Did you have something in your records that says you had it? Did you have a profile? Because if the answer is no, then you can't prove you failed because of it, especially if you passed an APFT before.

My nephew was diagnosed with asthma and was discharged with a medical and he was Army. If you were seeing a professional about the issues with your breathing (if this is the case), you should have gotten a waiver due to tests pertaining to your breathing; that should have given you the OK to pass on the PT test, until a said diagnosis was established.
Medical waivers do wonders.