When plugged in and you see 13.3v at the converter output, you should also have about 13.3v at the battery posts. If only 12v there, check your "battery fuse" or DC circuit breaker near the batts somewhere. If it is open, the alternator charging won't reach the batteries either.
However, if your batts die after running things for a while, that means that battery fuse is ok since the 12v from the batts can get to the lights and fans etc. Some other "open" would be between the converter and the battery bank.
You are getting 12v in the house with alternator running so it is not the DC CB by the isolator, but could be the DC CB at the other end of the wire from the isolator near the batteries (which might be in addition to the "battery fuse")
Some MH have some of this in a compartment next to the battery location instead of at the engine battery and house battery locations. I don't have that so others can help there--also with whatever a BIRD is, if it is not the same thing.
You need the PD booster control if not fitted so you can put 14.4v to those 6s to charge them properly.