To clarify, when my grid is up, my backup circuits are powered and grounded through the grid as required by code.
When my grid fails, my backup circuits are powered and grounded by the inverters ground coming from the 3 prong plug on the inverter itself.
All 3 neutral, hot and ground connections break before they make from one power source to the other.
This is simply the correct way to wire a backup system, safe for me and my equipment and safe for the service personell attempting to bring the grid back online after a failure.
Your inverter has 3 prong grounded outlets on the front of it, use the ground from the inverter ONLY!!! when running on inverter power and use the grids ground ONLY!!! when you are powered by the grid. DO NOT!!! leave both grounds connected together at the same time and you will be grounded properly at all times, which will keep the magic smoke inside your inverter.
Some people choose to do this by mounting a plug on the wall next to the breaker box connected directly to a breaker, they pull the the romex out of the box that was connected to the breaker and put a 3 prong plug on the end. Then they plug this circuit into the inverter when the power fails, and plug it back to the plug connected to the breaker when the grid returns.
This breaks one circuit completely then makes the other completely by swapping outlets manually, this is exactly what I do with relays so there is no manual human intervention required and the loads stay powered up at all times.
This is the correct way to wire an AC backup system whether you are using an inverter or a gas genny. BREAK ALL on one source before you MAKE ALL on the other source.
As far as grounding the inverter case (larger inverters above 1200 watts will have a ground screw), I leave that alone and don't ground it at all...it is widely viewed as it should be connected to the negative battery terminal on your bank but I have never had a problem not doing this. This ground terminal is not to be connected to earth ground connected to your grid.
Hope you understand....