When you are plugged in, the BIRD function within the BCC will cause the isolator relay (same solenoid relay is used to cross charge battery banks and function as aux start relay) This would bring the chassis battery online and start to charge it to converter voltage level.
When you unplug from shore power, it would take a minute or two for the chassis battery to drop in voltage and let the isolator relay drop out. That's probably the click you are hearing and accounts for the resultant loss in voltage after a short delay.
All the above would take place even if you had NO house batteries properly installed, which you apparently don't.
Your problem is right there at the batteries......
You might try the plugged/unplugged scenerio again with the CHASSIS BATTERY DISCONNECT activated. See if you get 12V at the house batteries when plugged in, and see if it still stays for a minute or two after unplugging. I'll bet your symptoms change.
