Looks to me like a factory made horn, from the second half of the 18thc. Actually looks pretty late to me. The paper work with it, from what I can see says he carved it, I take that to mean he did not make it. It also says his son, Bill, carried it for some time and did some carving on it, hence the name Bill on it. To me the collar looks pretty plain and that it is threaded instead of the horn, would be nice to see the tip.
The provenance with it looks pretty good but once again you can't see it all. If it were a Will, a list of property that was recorded, reference made to in in a diary, etc... it would help.
Try Zooming in on the paper work, especially the page on the left that is where you will see the reference to Bill and a date trail. It is possible that the story is correct and it would be a neat horn to have, as always; Caveat Emptor.
Link to Billy:
https://en.wikipedia.org/wiki/William_S._Hamilton From what I can find the Mrs. Alexander Hamilton, Angelia Livingston, referred to in the date trail is the wife of Alexander Hamilton, his father James Alexander Hamilton and one of Hamilton's, 1755-1804 Grandsons.
Tim C.