Is your inverter/charger functioning? If the coach is plugged into shore power or if the generator is running, then the inverter/charger should be charging both the house and chassis batteries in that order.
Does your coach have solar panels on the roof? if so, then the solar panels should be maintaining the charge in your batteries if the coach is outside in the sun while no electronic devices are being used.
If your batteries are down some, then don't start the engine to use the alternator to charge both battery banks. A heavy charging load will overheat the alternator and cause its demise. Use either the generator or shore power to first charge the house batteries to 12.5v or higher.
Your coach should have a crossover charging device - an Echo Charger or BIRD System - that will allow the charging current to be connected to the two battery banks in either direction. If the engine/alternator is on, then the chassis batteries will be charged first followed by the house batteries. If the generator or shore power is on, then the house batteries will be charged first followed by the chassis batteries.
Just don't run the engine to charge both battery banks if both battery banks are low. A new alternator will cost upwards of $500 installed.
The inverter, if on, is a drain on the batteries. The engine brain (the ECM) is a drain on the batteries. Without any charging - either solar, generator, or shore power - these "drains" will kill the batteries within several days to over a week.