First Power TO the inverter/converter likely comes from a breaker, often a 30 amp one, in your main breaker panel (that is 120 volt power) 12 volt power comes from a rather large fuse called a "T" fuse.
The GFCI likely get's its power from the Inverter so you have one problem, not two.
Some inverters (mine for example)( OFF is OFF and I mean OFF, no power passes through the box. Standby, on the other hand, lets me use this computer off shore power.
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times