I finally found this song in print!

From Holstein-Friesian World, Vol. 18, No. 18 (Laconia, NY and Syracuse, NY: Holstein-Friesian World, Inc., April 30, 1921), page 95:

THE MILK DERBY
Song by Kirkbride
(Tune—"Reuben, Reuben, I've Been Thinking")

I had a horse and his name was Napoleon,
All on account of his bony parts.
He was sired by a Hamiltonean [sic],
Cussed by everyone 'round these parts.
He was so thin you could see right through him,
Hair was as fine, as fine as silk.
Hitched him up to an old milk wagon,
Taught him to stop when I hollered "Milk."

As I was going to town one morning,
I met Ed Scott in a rig so fine.
Says he, "I'll race your old Hamiltonean."
Says I, "All right," so we fell in line.
Down the road we helter-skeltered.
We were going at a hang of a pace.
The old humdinger was coming out winner,
When Ed hollers "Milk" and I lost that race.