With a battery for the TT and no converter you should still be able to run the furnace and the other 12v things and being on shore power with 120v, the coffee pot and other 120v things too with no GFCI issues.
That's if the converter was the guilty party.
Not clear how you can measure zero output from the converter "pos and neg" but do get 20v DC across "pos and ground" The converter is "negative grounded" so that should not be possible unless there is a mix-up in terminology.
The converter 12v is grounded to the TT frame and so is the 120v shore power input. IE the 120v and the 12v share the same frame "ground" (It is not "Earth ground" but never mind that)
There is no connection between "positive" 12v and "hot" 120v but they share the frame "ground" BUT the 120v is between "hot" and "neutral" and 12v has no "neutral".
So how did the 120v GFCIs fry?