The COACH battery(s) provide the 12-VDC System inside your RV. They power your interior lights, fridge control circuit board, your Alarms, your temperature control circuit board and other things.
The battery is kept charged by the built-in Converter(charger) that operates on 110-VAC (external power) and when the voltage is actually at 12.5 VDC or lower the battery is NOT charged. You should measure around 14.5 VDC or so, if the charger is running and around 13.5 VDC if it is charged.
You should check that the battery is being charged first, then if it is but the voltage is LOW as I mentioned, check the battery for water and check the date on it. It could just be ready for replacement.
PS. - The engine running should have nothing to do with your interior lights operation or the charging of your COACH battery.