I found a tune dated circa 1820 (Work and Sing by Ronald Cohen, preface). There was no tune with it, unfortunately. Perhaps this was one of the copies of the song supposedly thrown off the floats at the festival which many sources reference.

The closest to an actual answer I've found is from Elbridge Brooks biography of Franklin (pp. 228-229), where he wrote the following: "When the States had adopted the Constitution, there was a great celebration in Philadelphia in honor of the event. There was a procession, a banquet, and speeches; and Franklin did his share in making it remembered. In the procession all the trades were represented at their work; the printers had a wagon on which was a press, and they printed and scattered among the crowd a song written by Franklin in honor of the trades. It was not much of a song or much of a poem. Franklin himself, you know, laughed at his so-called poetry. But it was verse; and, as a song, it became as popular as anything a real poet could have written.

Sadly, I've found no indication of a true tune, though I think it can be read to Dylan's The Times They Are A-Changing

(As a final PS note, one newspaper I found clarified that the "Happy and free, happy and free / Yet all are contented and happy and free" appended to the end of the first stanza was actually a chorus that was repeated after every stanza.)