100 feet is very long for an extension cord and long term, this isn't a realistic solution. I recommend having an electrician install an outdoor rated (NEMA 3R or 4) level 2 charging station. It will set you back a couple hundred bucks but will be much safer, look neater and charge the car in 2 hours.
The cord we bought has a GFCI breaker built in, but it's not working. It's constantly tripping. After 15-20 seconds it pops.
The station itself is anywhere from $400-$800 (give or take) and the installation cost varies depending on individual circumstances but the total cost in many cases is under $2,000.
The cord ran out air compressor and didn't trip. Whether the car's charger works I don't know since the inside of our garage doesn't have an outlet with a GFCI.
Have you ever successfully charged the car anywhere? Remember, this is a $30,000 car with sensitive electronics. The air compressor will tolerate fluctuations in the electrical current better than the car will.
Well, I can tell you this. I run my Toyota-provided EVSE on 35 feet of an originally 50 foot 12 AWG extension cord, itself plugged into a GFCI-protected outlet. Never had a trip. At work I plug the EVSE directly into a GFCI outlet. Never had a trip. You might have a defective EVSE but to find out, you need to plug it directly into a GFCI-protected outlet somewhere else to see.
But the problem isn't the EVSE, it's the cord's GFCI. The compressor could trip it just as easy as the EVSE if the cord itself is the problem.
Here's what I'd do. I'd return the GFCI-protected cord you bought and buy a plain cord and a GFCI outlet. I'd replace the outlet you are using with the GFCI outlet (shut off the breaker first unless you're used to working stuff hot, which it sounds like you aren't).
My cost was around $400 for an electrician to install a 240V outlet in my garage. A rebate from my electric utility paid for all but about $100 for the charging station itself. So about $500 out of pocket total.