I believe you have a CONverter, not an INverter. A CONverter produces 12V power from 120VAC and includes a battery charger. An INverter produces 120VAC from 12V battery power and is used tp power AC devices when shore power is not available. Few RVs of your vintage came with inverters, though someone may have added one later. Some inverters include battery chargers and others do not.
Is the thingy in the cigarette lighter an inverter that you can plug a AC-powered device into or just a voltage monitor, displaying the chassis battery voltage? In any case, it is displaying the chassis battery voltage, the battery used to start the engine. You probably have another battery (or batteries) that supply power to the house 12V system, e.g. lights, fridge circuit board, etc.. Your chassis battery may be fine but still no 12V power elsewhere, since the two systems are separate (isolated from each other). And is is possible that your engine alternator does not charge the house batteries, either by design or some problem with the isolator relay.