Have you measured the voltage at the battery terminals? Since your generator still starts, even when the dash display says 0 volts, I'd suspect the display, not the batteries. What you describe could just be the charging voltage from the alternator. The two battery groups are usually called the chassis battery(ies) and the house battery(ies). They are electrically separate from each other until a charging voltage (+13.4v) is detected, then the battery groups connect so both can be charged. This is done by a battery isolation relay, it often has a built in delay that doesn't connect as soon as the engine starts or converter/charger is plugged in. It also may delay disconnect when charging current isn't detected anymore.
On your dashboard, is there a battery boost, or emergency start switch that manually connects the two battery groups together, sort of a built in jumper cables? What does the dash meter show when it's pressed and held?
The chassis battery energizes all the functions for driving, wipers, turn signals, dash HVAC, head and tail lights, etc. The house battery powers all the living functions, refrigerator, interior lights, water pump, furnace, water heater, etc.