The inverter (if any) wouldn't be the problem, but the converter/charger might.
Terry's suggestion about the mysterious store/use switch is the first/easiest thing to check, but it sounds from your description like the 12v
lights work for a while but run the battery down, and I don't think those lights work off the battery in the Store position.
Check to make sure the 120v outlet that your converter/charger plugs into is hot when you're on shore power (check the 120v breakers if it's not.) Then test the converter itself... make sure it outputs DC when it has power, and check its fuses if it doesn't.