I think if Q was just working for CM the FBI NSA CIA and SS would have shut it down. I'm 100% sure Trump would have tweeted about the larp. He is not one to share the spotlight with rouge actors out of his control.
Impersonating Federal employees or MI is not legal.
Especially after the Avenatti photo; Q would have been investigated, and to the best of my knowledge no one has even been asked a question by the authorities.