I doubt the system would have been designed to drain the chassis battery with the generator running. As others have mentioned, you have something not functioning properly.
My generator starts from the house batteries. But my chassis battery was draining due to parasitic draw while stored. It's plugged in 24x7 so I installed a Xantrex Echo Charger:
Echo ChargerIt's about $30 more than the Amp-L-Start and frankly I don't know whether it's any better. My Trik-L-Start failed after a year and the manufacturer had a pretty crappy attitude about it. I didn't want to buy anything else from them so I got the Echo. It's been working flawlessly. Chassis battery is always fully charged.
