From The Knickerbocker, Vol. 31, No. 4, April, 1848, p. 375 [in a rambling column called “Editor’s Table”]:

The death of eminent men used formerly to give rise to more ‘tributes’ in verse than are common now-a-days. Among the objurgatory poetry elicited by the wilful murder of General Alexander Hamilton by Colonel Aaron Burr, was the following:

‘Oh! Aaron Burr, what have you done?
You've shot great General Hamilton;
You got behind a bunch of thistles,
And shot him dead with two hoss-pistils!’

There is a similar quote in:

Holden's Dollar Magazine, Vol. 2, No. 2, August, 1848, p. 508-9.

This is apparently just a short quip, recitation, or joke, not a song, but it has been quoted in several books with more or less the same words, and no additional verses.