I have never had a issue with the inverter and GFCI tripping. I have read in the manual of my Kisea Inverter that GFCI work the same with inverter as with shore power. One exception , testing a GFCI while the inverter is in Idle Mode can damage the circuitry of the GFCI.
This is from the manual
A GFCI has a test button on it that simulates a ground fault condition to test that the GFCI functions properly. A reset button is also present. A GFCI should be tested only when the inverter is inverting or transferring shore or generator power through to the GFCI outlet. It should not be tested when the inverter is in idle mode. If the test button is pressed while the inverter is in idle mode, the GFCI will appear to fail the test. The circuitry inside the GFCI may be damaged by the sense pulses that are generated by the inverter during idle mode.
As others have mentioned confirm the inverter is inverting power. The AC volt gauge should read 120V (If Pure sine wave inverter) usually the inverter will invert DC to AC with battery voltage 11.5V +. Turn on a electrical component (Microwave) to confirm there is out-put to power the microwave. If so try reset again.