One of our cats is a mouth breather sometimes. In his case, it's that he gets mad seasonal allergies and if he gets a boog, he'll mouth breathe till he can get it out- usually by sneezing 30+ times in a row- and then I have to clean up the nastiest hocking booger ever. He is our "problem cat" as he is also prone to UTI's and has to be on special food. It took some time to figure it all out, but our vet is wonderful!
I wish I had some actual advice to give you. I would say that if you have a nagging sensation that something is being missed, then it probably is, and if it isn't it doesn't hurt to be 100% sure.
I'll also have to agree with shopping around for a vet. You wouldn't stay with a doctor you didn't like, would you? SO why stay with a vet that you don't?