I didn't see a description of what you are trying to run. I know that if I leave the refrigerator on AC overnight, it will draw my batteries down. I switch it to LP. If you leave it on AUTO, it will default to AC if the inverter is on.
The old trick of pulling fuses circuit by circuit to see what is drawing the current is still a good idea.
If you still suspect the batteries, disconnect them from each other and see if one goes down rapidly. If so, bad battery.
W8RLM - Bob
2006 Winnebago Journey, Model 39K, 2013 Dodge Dart