Jim, rather than add to an increasingly confused post I thought I'd write to you directly. You got part of my point - if the coach remains on shore power you never draw down the house batteries, but the chassis (or starter) batteries continue to get drawn down. In this case it should not matter because so long as your float voltage remains above 12.8V the BIRD/Big Boy will continue to provide charging voltage to the chassis batteries, and it will be adequate to stay ahead of the draws on the chassis side.
If you are not charging check what the voltage is on float charging and refer to the inverter manual. In my case it was a bad temp sensor at the house battery sending a false high temperature to the inverter, causing the inverter to drop the float voltage to 12.6V, which caused the BIRD to open the Big Boy and stop charging the chassis side.
When this happened my chassis batteries were discharged in about 48 hours. I didn't need a charger, I needed to disconnect my temp sensor.