Your AC thermostat gets it's power from the AC unit itself through a two wire connection to a reducing transformer on the 120 volt power input to the AC.
If you do not have power to the AC thermostat, you do not have 120 volt power to the AC unit.
Since this affects the AC unit, it has nothing to do with a GFI or one of the breakers on your inverter/charger. It is either a problem with the breaker on the shoe power pole, or your main breaker in your electrical panel. You have lost 1/2 of your input power and this could be from a bad shore power breaker, bad plug, bad cord, bad transfer switch, bad main breaker in your power panel.