va capone-
I suspect the batteries are correctly connected.
The 600W inverter supplies AC for your TVs. It does not charge anything.
The other charger charges your house batteries. There is a solenoid that connects the house and chassis batteries, under certain conditions, to allow the alternator to charge the house batteries, and to allow the house battery charger to charge the chassis batteries. The solenoid is part of a cross-charging system. Either some path between the cross-charging system and the chassis batteries is broken, or part of the cross-charging system is not working as designed. Both are common problems.
You can figure this out for yourself. You need a voltmeter. Do you have one? Can you read wiring diagrams?
For something to try immediately, find the chassis battery disconnect switch and make sure it is turned on. Your coach's owner's manual (link
here), pages 146-7, may help you find the chassis battery disconnect switch.