You have systems that I do not have but after reading what a BIRD is, I would start there. When either battery is above 13.3V the BIRD will close a relay tieing both battery together so they would charge. So I am thinking that the Bird is bad or you have a broken wire to/from the the BIRD.
First step would be to measure the alternator voltage going to the BIRD at the BIRD and then measure the voltage coming out of the Bird for house battery voltage and if the same as alternator voltage bird is OK. I would then check that the alternator voltage is at the house battery if not, bad wiring from the BIRD to the battery.
Dumb suggestion, you have checked the battery cables are tight on the house battery?
Steve and Adele
2014 Flagstaff T12RBST
2005 Toyota 4runner 4x4