In your PS statement, you verified that you do not have an ATS inline w/ the power cord, so we rule that out as the problem.
First of all, this is NOT normal. While plugged into shore power, everything should be powered, including the batteries being charged.
Heres my new guess,........
The problem lies in your converter,....the thing that powers the DC side or the RV, and charges the batts. You just think the batts are being charged when plugged into the generator, but in reality their not. If they were being charged when plugged into the generator, they would be charging when plugged into shore power also. So your converter is NOT charging, period.
The only charging your getting is from running the engine.....thru your alternator.
I'm betting your converter will need to be replaced.
