Weather it's car repairs, or home repairs, always check the simplest things first

I wish to heck you'd asked on here before you started throwing money at parts. I used to maintain/ repair/ install HVAC. Unless things have changed, a dedicated circuit for a permanent heating/ cooling appliance does not need GFC They are "permanent" and are supposed to be additionally grounded. If nat gas, then the gas piping is supposed to be bonded to a ground. (The meter is isolated by dielectric joints same as a water heater)