the Chassis battery(s) should really have no need for constant trickle charging, even when in storage for long periods, as it simply is a starting battery, not one that has, or should have, any type of 'phantom' unknown draws while just sitting.
What might the 'other' cable to this battery be for? Probably for the 'Aux' or 'Emergency Start' switch, to tie in the Chassis and House batteries when either are low, like a set of jumper cables.
If the batteries are falling, I imagine you have left something on, that draws from the Chassis battery(s), such as a radio or video system, or similar, that you sometimes want to use even when you don't have the engine cranked, or have to leave the key in the 'on' or 'accessory' position.
as for the Microwave - it does not use 12v power, so the 'Use/Store' switch actually has nothing to do with it, THOUGH, the switch probably DOES disable the INVERTER, which is what powers the microwave, a residential fridge, if you have one, and maybe other outlets within the coach.
The Use/Store switch simply is an easy way to remove all lights and other 12v house items from the batteries, but not necessarily every 12v device, as C02, Propane, and Smoke detectors will always be 'live', even with the switch in 'Store' position.
For solar, the panel usually attaches to a solar Controller, then directly to the battery(s), though I imagine that it's for the House battery(s), not the Chassis. And, whether the switch is in 'Store', or 'Use', won't impact that.
While this is generally how most RVs work, yours can have some 'extras' that may tilt the scales one way or the other, but most likely not too much. Some factories do things 'this' way, while others may do it 'that' way...
I suggest that you simply have a phantom draw on your Chassis battery(s). You could test this by removing the NEG cable from the Chassis battery(s), and check back several weeks and see if the level is any different. If not, you have a draw to find. If it is then lower, you may have a bad battery.
Or drive to most any parts store which can load test the battery(s)...
'14 Palazzo 33.3 bunkhouse 34' diesel
KingTailgater2 Dish HD, 100w/5a SOLAR, BlueOx
100,000+mi since '14 - US, Alaska, Canada.... formerly Georgia, then NorthCarolina, now Alabama : )