Like Scott, I also am not sure how the BB would cut off chassis power. You say it “kicks in” after a few minutes of shore power disconnect. Perhaps you are hearing some other device, like the transfer switch, though that’s normally within seconds, not minutes. Is the battery boost switch on your dash (armrest panel?) off? Perhaps the BB or BIRD are miswired?
Loss of chassis power to anything would send me first to the streetside forwardmost electrical bay, wherein I’d check the power busses; and then the solenoid’s posts with a voltmeter, initially with the ignition key off and then check again with it on. Given its reputation, the one thing I’d suspect is the ignition solenoid there. Some owners have replaced it with the wrong one, or remounted it’s diode backwards.
But you say you do have power to all chassis components while on shore power?
Joel