The solar, unless its factory installed and someone may know more definitively how its wired (though no telling what PO might have done)
no need to pull battery(s) to check. No idea what your controller has, meter or just lights. Disconnect shore, check battery voltage. Run a fan some lights etc for a while. Solar should go to charge mode. I would check voltage out put of controller, then check at battery-should see same voltage, turn off battery cut-off, voltage should drop to standing battery voltage (not show charge input) if solar is wired thru cut off. As most controllers have an off switch-I dont know why panels would be thru main cut off though.
As to the 3way refer, and the inverter Im not quite clear. It sounds as though the inverter (12v DC to 110 AC) is wired to run the refer AC receptacle? Not common but some folks do. If thats the case the refer would be set to AC. Also, if refer is plugged into receptacle thats powered by inverter means even on shore or generator, the inverter needs to be on, unless theres another actual AC receptacle.
To use on directly on 12v DC the inverter wouldn't be used. If refer is set to DC it would be drawing directly off the battery(s) not thru inverter. Your 3663 will draw a little over 17amps in DC mode. Most trucks wont supply that and still charge batteries, So even driving its slowly drawing down batteries. You have solar input helping-but it takes a lot of diligence to use the DC mode in factory (truck charge) configuration. IE turning OFF refer when stopped-period even a few minutes. Turning off least an hour or so before arriving destination allowing battery(s) to charge etc. That said, assuming you precool the refer before a trip as already mentioned it will stay cold(ish) minimum for several hours.
Because of your curiosity and 'want to know' I recommend buying (or find a Harbor Freight "free" coupon for their $5 one) an inexpensive digital multi meter if you dont have one. Set to 20+ on the DC volt scale it will tell you exactly what you want to know on the 12v circuits. Test light would only tell you power present not if 12.2v and dropping or 13.6 & rising.
Set to 200+ on the ACv scale you can test the receptacles/inverter output-though for most all you want to know if power present and not looking at actual voltage so a AC test light (or portable power tool etc) would be sufficient on the AC stuff.
