Steve, thanks for calling my attention to the song in A SAILOR'S SONGBAG by George Carey. I was just looking at that book a few days ago, but I didn't have this question in my head at the time. Carey doesn't venture an opinion on whether this is from oral or written tradition or whether it came from a fellow American prisoner or from an English source. However, it does suggest that this particular ballad, #112, "The Baffled Knight", was being sung in 1778. The version in the SONGBAG does look like a sung version.
If nothing else, this collection probably made its way back to America after the war, before the turn of the century, and would thus count as a documented version in America in the 1700s. I couldn't find an exact reference in Carey as to when Connor, the prisoner who collected these songs, returned to America. It just says that he was part of a prisoner exchange on June 14, 1779, and was taken to France.
And thanks for your other suggestions as well, Steve. I notice that "The Baffled Knight" shows up in the PIONEER SONGSTER as "Katie Mora" (p. 9). Unfortunately, the Google Books excerpt does not give very much of the book, and the only information I could get on the dating was from the subtitle "Texts of the Stevens-Douglass Manuscript of Western New York, 1841-1856." I will add this to my library search list.