As in the best tradition - borrow!"My Country Tis of Thee" is tune of "God Save the Queen."
With careful research, don't think it would be too hard to find probable/likely tunes in an area/time frame. How, establishing which is the MOST probable that may be harder. Hey, we're working with an oral history and tracing it back to a "snapshot" time frame.
On the other hand - it would be absolutely fascinating research.
There are amazing coincidences. I worked on the Recreation Board a few years ago with Irv Finston - who in college, carried the tape recorder (then in the "army portable"version - i.e. it had a handle and it weighed less than a truck) in the 30's(? college) for ballad collector. Reason I know this was because I was singing one of the Shaker while songs setting up for a meeting and he recognized it. (from Chicago Il). Elaine.