Re: Batteries ...
Your coach/house batteries are charged by the same converter that runs your inside 12 volt lamps and appliances. Sometimes the engine charging system will be routed to charge them also.
Generally, your generator will be using the coach/house batteries to start, but there is that path to the engine battery. So, your symptoms indicate that you are starting your generator by this extra path and that is the hard way; could even cause problems. (I would check for some overheated wiring.)
Specifically, your symptoms indicate that your coach/house batteries have gone to the happy hunting ground in the sky. They'll need replacing if they've been that way for more than a few days.
Also, definitely get that converter checked out. That may have been the original reason the batteries died.