When I used a CHE with both a dimmer and a tstat. Both worked fine for me.
Using a CHE I used the wire fixture and would attach the tstat probe to the most outer ring pointed inward so it was in the path of heat. I still use this technique with my RHPs by gluing the probe inside a cable strap that I glue on the outer rim of the RHP. Has worked great and gives a clean look.
I prefer this to the often suggested dangling method.