Omicron Zeta Songs suggests "Itsy Bitsy Spider," which seems to be a close relative of "Battle Hymn":

Hairy-Chested Men

Specially for football archrival University of Pennsylvania.

(sung to "Itsy-Bitsy Spider")
Harvard's run by Vassar, Vassar's run by Yale.
Yale's run by Wellesley, Wellesley's run by tail.
Princeton's for the pretty boys and drunkards go to Penn,
But far above Cayuga's a race of hairy men.


("?")
Oh we are the race of hairy chested men, hairy chested men, hairy chested men.
Oh we are the race of hairy chested men, and we are from Cornell.
We are from Cornell, we are from Cornell (Cornell!)
Oh we are the race of hairy chested men and we are from Cornell.


("Itsy-Bitsy Spider")
Don't send my son to Harvard, the dying mother said.
Don't send my son to Yale, I'd rather see him dead.
Send him off to Darmouth, or better yet, Cornell.
And as for Pennsylvania, I'll see him first in hell!


("Glory, Glory Hallelujah")
To hell, to hell with Pennsylvania,
To hell, to hell with Pennsylvania.
To hell, to hell with Pennsylvania.
To hell with U of P-- P-U!


We were only, only, foolin',
We were only, only, foolin',
We were only, only, foolin'--
The hell we were! P-U! P-U!