I have been reading "Barges" origin stories half of the night and the most credible origin so far is "The Barge Song" in the Max Hunter Folk Song Collection https://maxhunter.missouristate.edu/songinformation.aspx?ID=1346
Its a collection of 1600 Ozark Mountain folk songs that he was trying to preserve for posterity. The recording is from 1969, so after the song first started appearing in girl scout books, but the wording is different, and more old-fashioned somehow than the version I learned at camp. So I am thinking that it is pretty unlikely that Max Hunter accidentally recorded a song that his source (Clint Piney) learned at (girl) scout camp, given that he was trying to record old folk songs and that's what the rest of the collection seems to consist of.
My hypothesis: "Barges" is actually an old Appalachian/Ozark/Mountain folk song that some anonymous counselor introduced to the Girl Scouts and it got popular within the Girl Scouts and then the broader summer camp community.
To test my idea I have spent the last hour digging through other old folk music databases but no luck so far. If anyone has some spare time maybe there's a cross reference in some old book/collection about American (or English/Scottish/Irish/Canadian) folk music.