Re: Inverter not charging batteries
If the converter is charging the battery while on generator power, that proves out the electrical path from generator, through part of the transfer switch, to the circuit breaker panel, through the circuit breaker that powers the converter, through the converter, and on through the 12V mini-circuit breaker and buss bar finally to the battery.
If that same path doesn't work on shore power, the areas that are suspect are 1) the power pedestal, 2) your power cord, especially as it attaches to the RV, 3) the shore power input side of the transfer switch.
If you're losing one power leg on the shore power side, that would explain why the battery isn't getting charged, but you would also see some other devices or outlets that have lost power.
Let us know what you find.