To give an update to my poets theory earlier:

I skimmed through the beautiful complete collection of John Crowe Ransom's works at the library last night (link) and didn't see anything. I think we can officially cross him off the list of suspects. Shockingly, I don't think any of his poems actually dealt more than metaphorically with the Civil War.

I've also skimmed through several books of poetry by other members of the group. Robert Warren Penn dealt primarily with literary works, but did put out some poetry as well. Though several of his novels are set in Tennessee, very few of his poems seem to be on the subject.

I skimmed through every book of Allen Tate's I could find and turned up blank.

Merrill Moore can be almost instantly written off as he wrote sonnets almost exclusively.

Work of the comparatively minor members of the group has been nearly impossible to locate. I haven't been able to look at either of the novels William Ridley Wills put out, and other than a clearly-unrelated play about Greek mythology, I can't even find a list of works by Sidney M. Hirsch, Alec B. Stevenson, or William Frierson. The works of James Marshall Frank (link) don't look promising, as does a list of poems by Jesse Ely Wills (link). Though Walter Clyde Curry was described frequently as a poet, I haven't found any evidence his works were ever distributed (though he does have some clearly non-fiction contributions, see here). It may be worthwhile to investigate the 1922-25 poetry journal run by Stevenson, The Fugitive, however, it is not indexed or available digitally.

Donald Davidson is my favorite suspect, partially because he does have several poems written about the Civil War and its changes and impacts to Southern culture and society (see Lee in the Mountains), and experience making fake dialect songs (see Big Ballad Jamboree), but also because he wrote an entire two-part book about the Tennessee River. Though I doubt it's the write lead, I'm working towards reading his folk opera, Singin' Billy -- though it's set in South Carolina, I don't think it's unreasonable to imagine the song cropping up in it.