The following is related to the UK...230v! I assume it's similar in US for 110v.
By "ground" do you mean that there is an extra terminal on the body of the inverter as well as the two connections to the battery?
By "frame" do you mean the METAL chassis connected to the axle or a METAL frame for the tear itself?
In the SBMCC I belong to it's suggested that if you should aim to use double insulated appliances. Otherwise this should be connected and the frame should then be connected to earth, as in the land you are parked on and the item fed by a 3 wire cable and plug.
Thats the definition of "Belt and Braces"...If you don't do so you might need to Brace yourself for a Belt from the electrics
Also use flexible cable instead of single-strand home wiring - it doesn't break as easily under vibration.
SBMCC = Self Build Motor Caravan Club.