There's a version of this in the DT already. This one is from 1870. the version Levy has from 1864 is about the same as the one in the DT. To the familiar nursery rhyme tune.

LITTLE WEE DOG
Sep Winner. With additional verses by Barton Hill. 1870.

CHORUS: Oh where, and oh where, is my leetel wee dog?
Oh where, oh where can he be?
Mit his ears cut short and his tail cut long,
Oh where, oh where is he?

Sausage is good, and bologna also.
Oh where, oh where can he be?
Dey makes dem of horse and dey makes dem of dog,
And I fear dat dey makes dem of he.

Whenever I see a Bologna, I shtop
And I whistle dis bootiful air,
But the sausages never run out of the shop,
So I don't think my leetel dog's there.

But either in Camden or Manayank or New York,
Or else in my own Amsterdam,
My leetel dog's made into beef, or to pork,
Unless he is chicken und ham!

De reason I tink my leetel wee dog
To sausages must have been mince,
I had a bologna for dinner last week,
And by dam, I have growled ever since!

My leetel dog waggles his stump of a tail
Whenever he wishes for prog(?).
If the tail was the stongest, I makes no doubt,
The tail would waggle the dog.

Well, perhaps I have tasted my leetel wee dog
And swallowed his last vital spark;
His flesh it was sausage, his skin it is gloves,
But what have they done mit his bark?

My little dog's spirits, that never would flag,
Are turned into "spirits of WHINE."
His tail is preserved as the last of a wag,
And his bark is "sulphate of CANINE."

I would say this is "folk" -- since the chorus has become a standard children's song...