Chassis batteries 12.2 ignition off 11.9 ignition on.
Alternator is not putting out the correct voltage to charge the Chassis batteries.
House batteries 13.22. BIRD coach 13.2, ignition (with ignition off 0) (with ignition on 11.69.)
This sounds normal. Voltage at the "Ignition terminal/wire" on BIRD is low because the Chassis batteries are weak.
Solenoid 0 with ignition on or off.
You would only have power here if you press/hold the Battery Boost switch.
Bigboy relay left battery cable 11.8, right battery connection 13.22.
Since the BigBoy solenoid is not "closed" you are reading Chassis battery voltage on the left terminal (11.
and House batteries on the right terminal (13.22).
I think the problem is the melted terminal at the alternator. Your alternator is not charging.
From the picture it looks like a Leece Neville Alternator - can you confirm?
On the red wire where you are measuring the voltage, are you measuring the voltage on the side of the wire "coming from the wiring harness" or is it the side "coming off the alternator"?
I think the Red wire is the battery sensing wire and that is what tells the alternator to charge, and the white wire is from the ignition switch and is what engines the alternator. However the sensing wire is only needed (I think) on systems that have the Duvac style isolator. I don't believe it is needed on the BigBoy style setup. I'll research this some more. Maybe someone else knows for sure.
In the mean time, I would work on repairing that melted connection. Maybe you can cut the melted connector out and just splice the wires as needed back to the alternator terminals.