How big is the inverter
Less than 1000 watt is normally TV/Electronics only
2000 adds Microwave and possibly GFCI
Inverter may have one or two push to reset circuit breakers or
Switch to reset (Some of these the Tripped and OFF positions are identical. IE: Tripp Light (Also Onan Generators)
Or it feeds a "Sub" panel of breakers somewhere in the RV
THough in my case the GFCI goes through the inverter no matter if the inverter is powering them or not (it does pass through) so none of this would apply
IN your case I'm guessing there is a multi-pole Auto Transfer switch somewhere nad it's the stinker or one breaker on the inverter is.