No. Do not link the inverter chassis ground to the inverter's neg input. Going to the battery neg does that. The chassis ground should go to the frame independent from the battery's frame ground.
As suggested, you could connect to the metal casing of the power centre that has the 120v breaker panel in it. The shore power cord's bare copper ground wire is grounded to the frame from there so that makes a path for the chassis ground to the frame
You can just not use the chassis ground at all, since it is for reducing radio and TV interference.