Are you sure you have the inverter's neutral and ground wired correctly and not swapped up somewhere along the way?
To the best of my understanding, the ATS should switch fine with no ground connection at all (but of course a proper ground setup is an important safety item). It should not operate differently when they are bonded vs. when they are not bonded, at least as far as I can make out.
None of the electric panels in an RV should have ground and neutral bonded. When plugged into shore power, the bond occurs at the shore power main disconnect, whatever that is (often the power pedestal, if there is one, but it could also be an electrical panel or disconnect or something else upstream). A built-in generator in an RV, and I think probably an installed inverter, should bond them together at that source--which is why the neutral needs to get switched by the ATS along with the hot(s).