I believe I actually came across such a banjo at the 12th Fret in Toronto. I mentioned to Rick when we were there about a year ago, but I'm not sure he was convinced.

The instrument was an antique, I believe German made (but I could be wrong about that). It had a 5th string in the normal location - but it had 5 strings at the head and nut. It was not in playable condition, and no one in the store at that time knew much about it ... but I was convinced that the purpose was to add the lower string - giving you the instrument I described in my post above.

I tried to tune the instrument that way, but it was missing a string, and it was a bit too out of whack to actually play (action too low, strings buzzing on neck).