The shore cord plugs into an outlet in the RV that is connected to the generator
It's called a manual transfer switch, the person who drives the RV is the switch moving the cord from generator out let to park outlet and back
To the OP
I have to agree with the others the converter..which is your dc charger and power when connected to 120vac is not supplying enough power
Also from your statement " a stable 120v source" you are not in a park on a 30 amp RV power pedestal
It seems you are running everything on electric, since you did respond to questions regarding propane for the fridge and water heater
A 20 amp home duplex receptacle, it's not enough power to run everything on electric only ....something will suffer
Perhaps this is why the dealer could not find anything wrong
They did not power everything in the RV from a low voltage 20 amp duplex outlet
Really we can't help you if you don't answer the questions asked and only give generic answers
And No the batteries should not be running down, you should not need to start the engine, while parked, as long as the converter is working and the 120v is really 120 not 105
Like said before turn off the water heater, make sure the fridge control says you are on LP aka propane.. If the batteries don't charge there is a converter problem..The converter could be in the box behind your 12v fuses and 120v circuit breakers,
Btw your batteries should last all night at least 24 hours when fully charged...30 minutes engine run is not fully charged
