If the GFCI is tripping, there is current leakage somewhere to ground. Moisture on the connections/wring maybe? Watertight connectors might solve that. Moisture in the receptacle? Bypass the receptacle to test. Defective component maybe? Disconnect the heating elements one at a time to see if anything changes. Leakage in the thermostat maybe? Take the thermostat out of the circuit to test. Bypass the thermostat temporarily by connecting the Hot directly to the heating elements and see if it still trips. A break or cut in the insulation on a wire somewhere? If the circuit breaker is tripping instead of the GFCI, then something shorted out or overloaded. You said you had two. Are both doing the same thing? Lower probability of some of the items listed occurring on two separate units. Maybe GFCI is defective, but probably not. Install the copper ground for safety for sure. If there is current leakage (indicated by the gfci tripping), then there is a possible shock hazard.
Some items don't work well on GFCI. Freezer in a garage or on a porch is tough to do with a GFCI. A little leakage in the compressor will trip it.