Yes the three-light tester doesn't work with inverters or Honda gens etc. No problem. It is not a "quality issue," as such. don't get thrown off by that.
Chassis ground of the inverter, even though the "ground" is not "Earth," is intended to reduce interference with radio and TV signals.
Deck- mount converters also have a chassis ground that goes to the trailer frame that is not a true "Earth."
Usually the chassis ground AWG spec is not that fat, so #4 for that job is kind of weird.
The inverter situation varies. You get some that do not even have a chassis ground that work just fine with shore power plugged into them and the converter turned off.
Some have 120v receptacles that have a sort of GFCI arrangement for their three-prong receptacles. Some don't.
IMO ignore everything and just do it according to the manual for that inverter.
