As I said, I think more than one rat is in the woodpile. The old "Rogues' March" tunes, at least those I've found so far, don't match the tune for "The Worms..."
We have yet to find tunes for "Army Duff."
"Elephant's Walk" is late, but I haven't found it either.

The earliest score I have found is "The Hearse Song," from "Songs My Mother Never Taught Me," J. J. Niles, D. S. Moore and A. A. Wallgren, 1929, The Macaulay Co. Reproduced in Lomax, p. 556-557, American Ballads and Folk Songs, 1934, with score. "The bugs crawl in..." is in the 4th verse. A WWI age was suggested by Lomax.