The largest lesson in this is to have a functioning battery and a converter that is charging the battery. So far, I don't believe you've measured the converter's output (maybe I missed it).
To have the starting battery fail, as well, means that there is probably no isolation between house and chassis battery. Something is drawing current from that circuit and the fridge would be one. It needs 12V to operate. You also have other parasitic draws like alarms and circuit boards, radio memory, etc. Has anyone used the lights? Are they LED or incandescent?
If plugged into 120V shore power, you should have the converter charging the battery and more than enough 12V power.