Ignore the non-problem! :)
I had that same inverter and it is the same as the Honda gen for that ground business. My other MSW inverters show all three lights on when you use the three light tester. Just ignore all that, it means nothing for running 120v things in the RV.
If you have the gizmo that needs the fake bonding plug for the Honda, the inverter might not need it, but just do what it takes.
I can say with the PSW 1000 that I had one funny issue with its chassis ground connected and my battery disconnect switch on the neg side, that when the shore power cord was plugged into the inverter, the battery disconnect did not cut off the 12v to the rig anymore.
Turned out there was still a neg path through the inverter via the chassis ground to frame and the shore cord ground prong in the receptacle. I simply disconnected the inverter chassis ground. (it is just for reducing RF anyway) That didn't happen with the Vector inverter which has GFCI receptacles. All too complicated.