Thank you, Doug and Tina. The picture is coming together.
From my research: Claremont, New Hampshire, had a controversial preacher -- Church of England, born in Connecticut, ordained in England, and serving in Claremont. He was open about his loyalty, in part because he had taken an oath to pray for Parliament and the King upon his ordination. Like Benjamin, he had refused to sign the Oath of Allegiance.
The preacher's name was Renee Cossit. The Claremont, NH church where he preached is still standing.
I am cautious about making assumptions -- but if Benjamin Leete went to Claremont, this may be another piece of the puzzle.