GFCI PROBLEMSThere are several types of GFCI problems that will be discussed and a technical article from Heart is also presented below.
You pull into a campground on a sunny, hot and humid afternoon.   You hurry to plug into the campground receptacle so that the a/c can start cooling things off.   Then, shortly after you plug in, the campground GFCI blows and you have no shore power.   Now what?
There may be one of several things that is causing the GFCI to trip:
You may also have a problem with one of the GFCI's in your D tripping.   This is almost always due to a faulty appliance or a wire that has come loose in the D.   If it is the latter, it is VERY important that you get this fixed before using your that circuit on your D.   Very rarely is the problem a faulty GFCI.
Here is the technical article from Heart Interface:
SUBJECT: USING GFCI's WITH INVERTER / CHARGERS
AC INPUT SIDE OF THE INVERTER / CHARGER
When installing an inverter/charger, it is important that the AC output side of the inverter feeds its own hot and neutral buses.  These buses need to be isolated from the input MAIN hot and neutral buses.  All the loads that are fed by the inverter need to connect to these isolated hot and neutral busses.  Using one common ground bus however is acceptable.  In many cases where GFCIs trip, the installer had connected the inverter AC output neutral to the main neutral bus.  Since the inverter grounds its output neutral when it is OFF or inverting, If the output neutral is connected to the main neutral, the main neutral also gets grounded.  A GFCI will detect this condition and trip out before the inverter has a chance to disconnect ground from its output neutral when it transfers shore power through.
AC OUTPUT SIDE OF THE INVERTER / CHARGER
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 and the circuitry inside the GFCI could be damaged by the sense pulses that the inverter generates during idle mode.
Nuisance tripping would be defined as the GFCI holding OK on shore power but tripping when inverter power is present.  This can sometimes be caused by marginal leakage between neutral and ground within the GFCI protected circuit.  This marginal leakage is not enough to cause a shock hazard but may be enough to trip a GFCI when inverter power is present.  This marginal leakage is often caused by surge suppression circuitry in some types of electronic equipment that may be connected to the circuit.  This surge suppression circuitry sometimes includes capacitors between hot and ground and between neutral and ground.  On sine wave shore power, these capacitors will not couple enough energy to ground to trip a GFCI.  But with inverter power, the capacitors couple more energy due to the harmonics contained in the waveform and this can be enough to trip a GFCI.