Agree, 12.7 VDC at the chassis battery with engine running means an alternator/regulator/wiring problem.
Same for no change in voltage on house battery when generator is on and presumably powering the converter, charger or inverter/charger.
And, yes after you solve either of the above, you can bypass the defective solenoid to charge both banks. BUT, have you confirmed 12+ VDC to the signal terminal of the solenoid. If no, disconnect the POSITIVE signal wire from the signal terminal and take a small wire (will be supplying less than an amp) from the large lug that has 12.7 VDC to the positive signal terminal. If solenoid closes, you know the solenoid is OK. If not the solenoid is bad.
Safest way to bypass the solenoid: Mark (tape, paint, etc) all the wires on one large lug. Remove them and ADD them to the other large lug.
Be sure to return wires to their original position when finished.
Do NOT use a jumper wire to do this with engine running or driving, as the jumper could "jump off" and create a dead short directly from a battery.
