Time to break out a voltmeter.
Batteries are the heartbeat of the RV and until you get the heart running properly there is no reason to start guessing about bi-directional relays and other systems.
Smacking the inverter is not a great idea at this point without getting those measurements.
The charger should come on when connected to shoreline or generator. A voltmeter in the DC range across the batteries will tell you instantly if the charger is putting out anything. You should look for 13.5 up to 14. If anything lower, you must check the output right at the charger. Lots of things inline that could be causing issues.
Dead batteries can take overnight if they come alive at all and bad batteries may die quickly even after charging for hours.
Myron & Deborah
08 HR Endeavor 40 SKQ
400 Isl, 400 HP/// Now 450 HP
2016 Ford F150 Limited V6 with Ready Brake