I'm having real trouble scanning the words above to any version of "Soldier's Joy." I can sort of see doing the first line of each verse to the whole A part of the tune, and the repeat of A would be the second line of the verse, but the "I don't want no more ..." just doesn't come close to the B part of "Soldier's Joy." Not the way I play it, for sure.
I'd say that whatever "Payday" song fits "Soldier's Joy" isn't the same song as "I Don't Want No More Army Life." I have heard people say that "Soldier's Joy" -- the real "joy" -- is payday itself, but not in reference to a song.