A notice in the Evening Star (Washington, D.C.), June 22, 1863, p. 2, announces the performance of a "sensational drama" titled "Young America and Old Ireland, or The London Correspondent in Camp." It seems to include a song, "Paddy [sic] Murphy (of the Irish Brigade)," sung by George R. Edeson.

OTOH, "Paddy Murphy" may have been the protagonist of the play and Edeson the star. The anotice is entirely ambiguous.

"Paddy" or "Pat Murphy," of course, would be a stereotypical name for a member of the Irish Brigade. I haven't found the name of the playwright or any text of the play.

More apposite is a notice in the Nashville Daily Union (Sept. 27, 1863) announcing the performance of "a new song, 'Pat Murphy of Meagher's Brigade,' [sung] by Mr. Stewart."

Nashville, Tenn., was occupied by Union troops from 1862.

It's tempting to identify "Mr. Stewart" as the "Fatty Stewart" of the songster.
"Fatty Stewart" (J. C. Crossy, 1834-1905) was a well-known comic singer of the period.