Welcome to the forum.
If the RV is plugged into shore power and is actually getting shore power (circuit breaker didn't trip and you're actually not getting power to the RV) then there is probably an issue with the converter. The CONverter (different than an INverter) is going to distribute the A/C electricity as well as convert A/C to D/C for charging the batteries and powering other D/C powered items (lights, thermostat, refrigerator controls, etc.).
As mentioned the first thing to check is that you are actually getting shore power into the RV. If you are then check the converter D/C output voltage using that voltmeter you mentioned across the wires connected to the battery terminals. It should be ~13.6V. Since the batteries are dead it's likely you're not getting proper charging voltage but it should be checked anyway just to insure whether the converter is the problem or not.