OK, I've never owned a new house. Or a new car.
I think OP's inverter only has sockets for plugs. Could use:
Yellow Jacket 2879 14/3-Gauge SJTW GFCI with Lighted End, Yellow, 6-Foot. Rated for 125-volts 15-Amp. Manual test-reset buttons, clearly marked. Right angle plug converts outlet into GFCI protected receptacle.
www.homedepot.com
A portable GFCI like that would disconnect both line and neutral. An outlet or breaker GFCI would just disconnect line.
Line and neutral of house wiring are supposed to be electrically connected, at one point. Usually this is the main breaker panel, where grid power enters. For marine/RV with shore power plug, it is connected through shore power pedestal to that main panel. When unplugged from shore power, neutral is connected by relay to ground inside the inverter.
For a stand-alone inverter such as this one, with no AC input, it is often floating. Some inverters can tolerate having neutral wired to ground externally. Some can't, because they deliver 60/120V split-phase, with the center tap connected to the battery. Having 60Vrms relative to ground (85Vpeak) is less hazard than 120Vrms, 170Vpeak, but could still cause a harmful shock. A GFCI may note that it is wired incorrectly. Some will still function, but the "test" button may not (because test current leaking to ground through a resistor would be half as much.)
I suggest using a volt meter to measure AC outlet voltages (Line to neutral), (Line to ground), (neutral to ground), (line to battery negative). Tests with high impedance (a meter) aren't completely conclusive. Try also an incandescent bulb (to apply a load) across those pins, and measure voltage across the bulb. With inverter shut off, measure resistance between neutral and ground.
Maybe neutral to ground is already zero ohms, in which case nothing to do.
Maybe neutral to ground lights a bulb or applies voltage across it, in which case it can't be grounded.
Maybe line to battery lights a bulb or applies voltage across it, in which case grounding neutral could make battery wires electrically hot with AC.
Ideally you end up with neutral grounded, either inside the inverter or externally. Then it is like house wiring and GFCI work and self-test as intended. If you can't then you just have portable inverter generating 120V across its line/neutral; portable GFCI would be good to try. Grounding the ground pin is still good, so if you hold a tool or appliance its case is grounded and won't shock you.