The piece in Randolph is not the original, but dates from about 1912, according to Cray (p. 199-200), and was sung to "The Red River Valley." Randolph does not mention a tune nor a date other than 1934 when he collected it. Randolph-Legman also mention the tune Reuben, Reuben.
Many pieces of sheet music from Harrigan's comic "plays" are in Levy and American Memory, including "Squatter Sovereignty" but not "The Tramp's Lament." Randolph says "According to a writer in the New York Times Book Review (July 7, 1940, p. 19), this piece 'was among Edward Harrigan's songs from the play Squatter Sovereignty produced in 1882'." There is another song of that name (1878) by another composer, but it is not the same.

I have been unable so far to find the original.