Here's a very interesting read.
http://www.rvnetwork.com/index.php?showtopic=106689If I understand right, the grounding screw on the inverter is there for the AC ground, not the DC side of the system. So, if there is a fault and the AC ground wire is energized, and the screw is bonded to the trailer chassis, the trailer becomes energized with AC. If you happen to be standing in wet grass and touch the trailer frame... Zap! You become the fuse between the trailer and the earth. This is assuming the chassis has no earth ground of its own, and the trailer is using the inverter, not shore power. The point is also made that some inverters internally bond the "neutral" AC pin and the ground together, which can also create a charge differential between the trailer and the earth if the ground screw is bonded to the chassis (potential zap). I will need to use my multimeter and check for continuity to see if this is the case.
The conclusion in the 3-page thread above is to NOT connect the ground screw on the inverter to the chassis. Now in my case, there is a metal leg on the tongue that might facilitate an earth ground from the chassis, so maybe the ground connection from the screw to the chassis would work properly if the leg is deployed and the trailer is not hooked up to my TV and "floating."
My inverter works fine without connecting the ground screw - I hooked it up to the battery bank today. No part of the trailer's electrical system grounds to the trailer frame currently, so it is always safe to touch the metal floor. I am still on the fence about whether to connect the inverter's ground screw to the steel chassis. In either scenario, the GFCI should trip and save the day if anything went wrong.