I had tmj myself. I too have a bite plate appliance that i wear. I have very very intermittent problems with ear ringing... mine only rings, buzzes or rings for a few moments and then goes away. Maybe thats why i learned to live with it.
I don't have TMJ, but have had ear ringing for a long time, with Lyme. Mainly in one ear. It gets louder in the afternoons, and even louder in the evening. No idea why, but it's the least of my worries.
When I started taking Cat's Claw, I notice that whatever is stuffed up inside burbled a bit and seemed to open up some. In me, I think it's inflammation, but haven't pursued it with the MD.
Good luck! Let us know what you find out.