I had the same perplexing issue a couple of TTs ago. Exactly as you describe.
After pulling what's left of my hair out trying to figure it out, I decided to follow the 12V positive wire. I probed it at the hitch where it went under and into the TT. Power there, but nothing inside at the converter fuse panel.
I got on my back and slid under the front, behind the frame cross member where the wiring went in.
I shone a flashlight up to the bottom of the floor, right behind the front frame and discovered a cluster of connections hidden in there. One of them was an inline fuse for +12V. I opened the fuse holder and Bingo! There it was. The fuse was not blown, but the holder and ends of the barrel fuse were corroded, just enough to make the connection intermittent.
I cleaned up the connections, put it back together and voila! End of problem.
It never happened again.
With any luck, perhaps you can find a similar issue.
Try tracing your +12V feed from the batteries all the way back to the converter.
When plugged into shore power, the 12V comes from the converter, regardless of whether the battery works or not.
2007 GMC 3500 dually ext. cab 4X4 LBZ Dmax/Allison - 2007 Pacific Coachworks Tango 306RLSS
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