We started down this path for our 3yo (who is less anxious than your son), and during the peak period we decided to put his name down for a devt paed (at least 4 month wait for private then) on the off chance it waa autism etc, and decided to take him to a psych for the anxiety.
In the end, we abandoned it as our son's primary problem is socialising with peers (he doesn't like kindy, just plays by himself, scared of other kids). He just seems to be an introvert, so we are focussing on slowly building relationships with other kids.
But, in your case, I would go on the waiting list for the devt paed and start seeing a psych when you can.
Hugs. It's hard when you have an anxious child. It breaks my heart when my son pleads to stay home from kindy.