The Hound Dog Song ("You gotta quit kickin' my dog around")
My Canary Has Circles Under His Eyes
Doney Gal (a cowboy's love song to his horse)
The Preacher and the Bear
The Pig and the Inebriate
Fod (with a woodchuck, skunk, rattlesnake, and dog)

Maybe you could write some new words to "While Gamekeepers Lie Sleeping":
Oh, I've got a mutt, and a good mutt, too,
And I keep him in my keeping
For to chase the strays that yowl outside all night
While the wife and kids lie sleeping.
;-}