On my Sun Voyager charging works in this manner:
1)Engine Running : Chassis and Coach battery charged from engine Alternator.
2)Shore Power : Chassis and Coach battery charged from Xantex Inverter/charger.
3)Generator : Chassis and Coach battery are charged directly from generator.
Here are few caveats:
a) I have a control panel above driver seat for Xantex. If it is not set to charge, then batteries will not be charged by it. Sometimes when I am disconnected from shore power and start the engine up, the Xantex does not turn on automatically and I need to turn it on and set to charge.
b) Chassis and Coach disconnect switches next to door: If either is 'disconnected' they will not be charged. Obviously you have to consider things like you can't start the engine if Chassis was disconnected, but if one was to become disconnected after engine is started, it may not get charged.
c)inter-connect solenoid: If this is not working properly, then you might only get either Chassis or Coach charging (depending on which 1, 2, or 3 method above is being used) but not both. This is the solenoid one activates via dash switch to have the Coach battery help the Chassis battery start the engine. In some configurations it is also used to bridge the batteries together for charging.
d)Some coaches, particularly most newer ones have what is referred to as a BIRD (Bi-directional Isolator Relay Delay). This device 'inteligently' connects your House and Chassis to the charging source depending on the source and the charge state of the batteries. If you have one and it is faulty then you would have a problem.
There is a thread here from someone diagnosing a similar problem(maybe you
If I had to guess about your situation, when you are on shore power, your chassis battery is disconnected so it is not draining. Then your Inverter is not set to charge, so your House batteries are slowly draining whenever you use a DC voltage device (most RV lights), but your AC voltage items (fridge, microwave, tv) are being powered by the shore power. Curious though how your chassis battery is being isolated.
I hope this helps.