Welcome, and an interesting discussion. First I would remove "Generator" as a source, since the source of the AC doesn't matter to the pod. If you argue for keeping the generator, you should add a new source, "External battery charger".
Stereo, lights, fan, water pump, some models of tv, the tv antenna signal booster and LP detector always operate from 12 volts DC. While the 12 volt source is the converter - fed by 120 volts AC - when the pod is plugged in, I don't think it is accurate to connect "water pump" to "AC", for instance. It is more accurate to connect the output of the converter to the DC system, removing the line from Shore Electric to the Battery. The converter does 3 things at one time - it is a regulated battery charger, 12 volt DC source, and it is the pod's utility panel, with breakers, passing the AC connection through to the other appliances without modifying it in any way.
The furnace burns LPG but uses the DC system for the thermostat and ignitor.
Microwave, A/C, outlets, some models of tv only run from AC (connected to the converter, but again, it doesn't do anything to the AC supply other than cut it off if a breaker trips).
Stove is exclusively LPG, gotta light it with a match or lighter, although it wouldn't be too hard to add an ignition system if you really wanted it.
Refrigerator and water heater use all 3, depending. The fridge's control board runs from 12 volts, the LPG-mode ignitor system, as well as the DC-mode heating element. There is also the AC-mode heating element, and finally the LPG-mode flame. The water heater has two separate systems, LPG and AC, but uses DC too. The LPG system has a DC ignitor system, and the AC system only uses AC.
Smoke detector and CO detector have their own 9 volt batteries.