I can't believe others are doing the same search. My mom has broken her hip so with her help (she has dementia but it was amazing how she could remember this song) I am trying to resurrect the old song. here is what I have (plus the couple of lines I was missing that I got from your post)

I had a horse and his name was Napoleon
named on account of his bony parts
he was sired by old Namatoneon
and damned by everyone who rode in the park

he was so thin you could see right through him
hair was as fine as the finest silk
I drove him in my milk wagon
and when I wanted to stop I hollered out Milk

One day while driving through the city
along came a rig with a jay so neat
he said mister I will race you
round the block and along the main street

off we were riding helter skelter
we were a'going at a terrible pace
sure as I'm a sinner I was coming in the winner
when he hollered out milk and I lost that race