I never wire a ref to a GFCI. I use GFCI in kitchen, garage, basement, outside, any place they will get along with motors.
What amazes me is that I have 2 wells wired to a GFCI. One well the pump is above ground and the other the pump is in the well. Both pumps get along with the GFCI.
You may be wondering why I have them wired up to a GFCI? I'm using the GFCI as a disconnect. I have a water alarm that shorts ground to common if it detects water on basement floor. Shorting ground to common on the GFCI will trip it, disconnecting power to the pump. It really works, prevented my basement from a flood when the water softner decided to spring a leak.
Back to the AC problem. #1 Why not remove the power cord, install an old power without protection, connect it to a grounded outlet and see what happens? #2 Or plug in the old style 3 prong cord from A/C in to an extension cord with a GFCI?
I would do #1 first while measuring the current on both the power and ground wire. I like to live on the edge. Love to see sparks fly, don't you? Stimulates the heart. If A/c works, then move to #2.