1. All outlets you use on the Nissan (Panasonic) EVSE, whether or not it's been upgraded by us must be grounded. If you plug the unit into a non-grounded outlet, the green "Ready" light on the EVSE will flash and it will not enable charging.
2. Many existing GFCI's can nuisance-trip when charging with the EVSE. This is due to a slight amount of leakage current in the car, and is not harmful, but obviously frustrating.
3. The EVSE already includes GFCI detection that is designed to avoid nuisance-tripping while charging. This means you do not need an external GFCI to preserve safety.
If you are temporarily charging, try to locate another outlet and try it. If you are attempting the use the location for ongoing charging, I'd either replace the GFCI with a newer one that's less sensitive, or (if code allows) remove it. You could install an additional outlet next to it that is not connected to the GFCI and only use it for the EVSE. If you have an upgraded EVSE, simply install an L6-20 outlet (even if connected only to 120v), which will prevent someone from accidentally using it and getting hurt should there be a ground fault.
Leviton is specifically marketing a GFCI outlet supposedly compatible with EVSE's. It's called the Evr-Green GFCI, and it even includes a light for nighttime use.
You do not need bother with an outlet tester, as the EVSE already tests the outlet when you plug it in.
If it is the extension cord that is tripping the GFCI, then it will likely trip even when the EVSE is not plugged in.