What did your AME say, he should have some idea of the timeframe.
This sort of thing might take north of six months, so you might need to talk to an aviation medical consultant. AMAS has a good reputation, not sure if they can expedite but they would certainly have an idea as to timeframe, and also know the right people to prod.
I'd get right on it, whatever they charge is a pittance compared to many months off the top end of your career (or losing the job altogether).