Joe, your Billy Barlow (2) of 1836 is the one performed by Jack Reeve but is missing three verses in the sheet music published by George Endicot, New York.

Verse 5
I went to the races on Long Island so gay,
The man at the gate he ask'd to pay;
What pay, says I, and I looked at him so,
Pass on Sir, I know you, you're Mr. Barlow.

Verse 6
I had been on the track, but a minute or two,
Before the people flocked round me, what I tell you is true;
Whose that little fat Gentleman, does any one know,
Yes says a young lady, that's Mr. Billy Barlow.

Verse 7
There's a nigger been here, who they say was Jim Crow,
But he cleared out the moment I came, you must know;
If you doubt what I say, I can prove it is so,
Just look at the rigging of Billy Barlow.

Verses 8 and 9 as given.