I have a 2012 Winnebago Itasca. I bought all new Interstate battery in February, took the coach to Florida for 7 weeks without battery issues. All seven weeks we were plugged into shore power with ...
7 weeks with the chassis battery switch off, followed by dead batteries after repairs, followed by dead batteries after 3 weeks can suggest:
1. Your new batteries are toast or at the very least compromised. 2. Your charger is not working and/or you have blown fuse or other wiring related issues.
I would not just assume it's the charger (you said inverter). A voltmeter would help as would load testing the batteries. Try charging the batteries with a 20A auto charger could shed some light...