(Note that this is written from Britain where the wiring and rules are different, although the physics of how electricity works is not yet under government control in any country.)
A GFI (alias RCD) trips if the total leakage current goes over its threshold for its detection time. Because every device has a small leakage current, it's possible that the car and the EVSE both contribute to the safety trip. Example: if the GFI limit is a nominal 30 mA but it's drifted a little out of spec to 25 mA, the EVSE has a 7 mA leakage and the car has 20 mA ... total 27 mA, more than 25, click, power off. Neither car nor EVSE alone would cause any problem. (This is a growing issue in Britain where a lot of small electrical devices have a milliamp or two leakage, and older homes have a single 30 mA RCD protecting a whole house, or one floor of a larger home. Add together laundry, kitchen, network and other gadgets and the total is close to the RCD/GFI limit, and one more device causes a "nuisance" trip.)
Leakage currents can be independently tested by an electrician with the right measurement equipment and skills. Lesser skill and simpler equipment should locate any incorrect house wiring -- sounds as though that at least might be a useful check here.
If the car has an insulation fault on one side only of the two type-1 connector power leads, the leakage current will be much higher for 120 V charging with the plug one way round than the other; and for 240 volt charging, where both connectors are at 120 V from ground, there will be high leakage whichever way it's connected. A wiring fault in the car, or a fault in its on-board charger (OBC), are both of them possibilities, and both repairable, so don't either rule those out or panic if they prove to be the cause.
