Use separate circuit, mount a new outlet
If you insist on using the original outlet
You will need a 'triple pole double throw' switch
Switch all three wires of each circuit
The outlet will be on the OEM circuit or the inverter
No way to accidentally connect then together, or make something electrically hot that should not be
You will need to mount this switch at/near the outlet, your looking at two Romex lines into the switch and one out to the outlet
IMO... The best way is a new circuit, Romex or extension cord to a new outlet, plug everything into the new outlet run from inverter full time
If when shore power or generator is in use, your converter will easily keep up and keep the batteries charged, no reason to swap back and forth
All my electronics are on inverter full time