Sorry there is so little information on the wiring for your Palm Breeze. The same is unfortunately true for all the GS products, as the factory has some sort of wild hair about them being "trade secrets" or something.
Yes, the inverter can drain the batteries in short order if large AC loads are left on and powered by the inverter's output. I would have thought that the coach and chassis switches in the stairwell would shut that inverter off, however, along with all other DC loads. Something is not quite right...
If you have a clamp on DC ammeter you can look around for how much current is being drawn from your coach and house batteries and maybe trece it down to the offending load. Lots of inexpensive meters have clamp on AC capabilities, but a few also read DC amps as well. The one I have has two DC ranges, 40 amps and 400 amps. You can get one at Sears for about $60 (careful to get the right one, the very similar model that reads only AC amps is $40...)
http://www.sears.com/shc/s/p_10153_1260 ... 482369000P