Larry, at 13.4v your Bidirectional Isolator Relay Delay (BIRD) switches charging to the other battery bank. When on the road, it switches from an alternator-charged chassis bank to begin charging the house bank, and when plugged in it switches from an inverter-charged or generator-charged house bank to begin charging the chassis side.
The BIRD has been known to fail from corroded or loose high resistance connections. But as reported on this Forum recently, the Big Boy solenoid that does the actual circuit switching can fail from corroded internal contacts. When on the road have you noticed whether or not your house batteries arrive fully charged? If not, the Big Boy is suspect. It could also be your Magnum inverter’s charger has failed, but not likely since your house set is getting charged when plugged in... and over time it goes from Bulk Charge to Absorb to Float Charge, right?
Use a voltmeter to see if when plugged in to 110v power and the remote reads Float Charge (at least 13.4 volts on the screen) that both large wire lugs on either side of the Big Boy indicate identical voltage readings. Over your battery tray and between the chassis and coach Main Switches is a gray door behind which are both above mentioned devices, large-amp 12v fuses, and other components. The Big Boy is front and center. The BIRD is mounted on the inside of the door.
Others here have disassembled, cleaned, and reassembled their Big Boy. Search the Forum.
I’ll leave generator commentary to others more familiar with that circuitry, except to say the genset may rely on fully charged chassis bats to stay running, and they aren’t at that point except via the alternator. With a working BIRD/Big Boy, the genset would eventually go from charging the house set to doing the chassis set, thereby then supporting the generator.
Joel