So, you have 12v when you are pressing the aux start switch and when the engine is running. The aux start switch manually closes a relay or solenoid, generally located in the chassis-to-coach power interconnect box. There are a number of styles of these boxes that use a number of relays, remote breakers, and solenoids to automatically and manually connect and disconnect the chassis electrical system from the coach (house) electrical system. The purpose of these systems is to allow the chassis power system to charge the house batteries and provide 12v power to house 12v uses while you are driving down the road. Generally speaking the chassis-to-coach interconnect system engages when you turn the ignition on and disconnects when you turn the engine off. This is no big deal because it is pretty difficult to overcharge a 12v lead-acid battery pack on an alternator that is outputing somewhere around 14v.
If your house (coach) had been converted to lithium batteries you would need a DC to DC charge controller to moderate the power draw necessary to charge/maintain lithium batteries from the chassis alternator. You'll burn out a regular chassis alternator if you just swap in lithium batteries in exchange for your lead acid because the power demand from the batteries will out strip the alternator's ability to charge them, there will be constant demand for higher voltages from the alternator than it is designed to provide. But this is beside the point, but food for thought.
Going back to the behavior you experience, when the engine is on your house 12v system is actually being powered by the engine electrical system, not your house batteries. When the engine is off you have no power to your house 12v, meaning that the circuit to the house batteries is open, but when you press the Aux switch you have power. Again, when the aux switch is pressed that is connecting your house 12v electrical to your vehicle 12v system.
I believe something has to have broken in that interconnect system when you last used the aux start switch. My money is on the solenoid or one of the solenoid pairs (I'm not sure if your system has one or 2 solenoids or how they are controlled, some are electronic and some are just electromechanical) that is closing the circuit when you press the aux start button, but is not closing the circuit to allow the 12v coach system to be energized by house batteries.
I have an old Bounder and my RV chassis electrical box is under the "hood" at the front of the vehicle, basically right in front of the driver. If you have a Class A, I would hope that this box would be near the front of the vehicle since the contents of the RV chassis electrical box, is engaged every time you turn on the engine.
One more thought, if the chassis interconnect solenoid isn't the issue, it may be a fusable link someplace, likely also near or going to/from the chassis electrical box that has been blown. You are looking for a fuse in-line with a main power line to/or from the chassis electrical box. Likely to the chassis electrical box from the house battery because that is where the "break" seems to be occurring. Because there is power to the house when the chassis is running, that would indicate that the connection from the chassis electrical box to the house electrical circuit is just fine.
You also state that when you are on shore power your house batteries are charging. Generally speaking the power converter, that converts AC to DC to charge the house batteries is directly connected to the house battery connection. In most situations if you are on shore power your converter could, although it is not recommended, power your house 12v circuit without the house batteries in place, however you'd have to remove the bad house batteries and complete the circuit that is made by the house battery for that to work. You could leave in place a marginal house battery and then let the converter carry the loads, but a good converter is always trying to maintain the house batteries and that may lead to "cooking" a bad house battery, which isn't a good thing either.
I'd also like to point out that you don't need to be running the generator to use aux start to "jump" your chassis. Ideally, if your house batteries are fully charged, they should have sufficient power to start your engine. This isn't something you want to do all the time because house batteries are deep cycle, not designed for bursts of power, but it is there for use in a pinch.
I carry a Noco GBX155 to do jump starts because using aux start on your house batteries is not something you should be doing regularly, plus these new jump boxes are just better at providing jump start power. Don't buy the knock off brands of jump boxes, most are junk. Noco is the only way to go. Oh, and if you do get one of the newer Noco jump boxes (basically any of the GBX series units), don't let them get fully discharged, they really don't like being discharged to 0, they don't let you charge them up from that point without a lot of pain and suffering. Rule of thumb is to do your jump, and then get the jump box plugged in to charge as soon as you can. It is also a lot easier to do jumps for folks that are stranded using a jump box because it is hand held, you don't have to maneuver vehicles to face each other and doesn't stress out a donor vehicle to jump the stranded vehicle.
