So since I have the same EVSE and I posted last night I have been doing some research to see what is going on. Everything I have found seems to point to the ground fault protection in this Clipper Creek being very sensitive to electrical noise, most the culprits I have found seem to be high drawing small appliances, keurigs, cuisinarts, blenders, etc.
The best solution I can find seems to be isolating it to its own circuit with a dedicated ground, not exactly a cheap option unless you already have a circuit setup for that.
You get 240V from two 120 V lines that are 180 degrees out of sync (think of it as +120 and -120, the difference is 240V).
A GFCI uses a torrid transformer installed on the "hot" wires to sense total current on the circuit. Any leakage to ground will cause the GFCI to trip.
The high current draw on one of the 120v legs is causing enough imbalance to trip the GFCI. It could be, as stated above, that the Clipper Creek is overly sensitive (some voltage drop is inevitable - ever notice the lights dim when you start a vacuum cleaner?).
It is also possible you have other wiring issues that lead to excess current drop. If all your "high load" 120v circuits are stacked on one leg, then that leg will be more susceptible to voltage drop.
In your breaker box, the legs are staggered - every other slot is on a different leg. A 240V breaker installed in the box will have one from each leg.
If all your heavy loads are biased toward one of the legs, moving a few breakers could help mitigate the issue. Current code require two separate "appliance" circuits in the kitchen (on opposing legs). They should be wired out so the physical plugs are used equally (if one circuit is only on the island and in the corners, it is likely to be used less).
The GFCI does not measure the ground or any "noise" on it. If the ground wire were run thru the torrid, any "leakage" to the ground would not trip the GFCI (the "sum" of voltages would stay the same). I doubt a new ground wire would solve your issue as it is not used by the GFCI.
It's possible to measure the voltage (and any drop) on the individual legs. You should be able to report those numbers to Clipper Creek and see if the GFCI is overly sensitive and needs to be upgraded.