GFCI's work by comparing current between the hot and neutral wires. If they differ by 4-6mA or more, that extra current is going somewhere other than through the circuit and is reaching a potentially dangerous level. Tesla EVSE's test presence of ground by intentionally "leaking" a couple mA through the ground pin. Normally this should be less than the trip threshold of everyday GFCI's. If everything's wired properly, your mobile connector may be leaking a little too much during this test.
Do you have any other known good GFCI-protected circuits/receptacles you can charge on to test?