TMac,
Three years ago I wrote to the town about the course. The historian, Doris Finch Watson, wrote back saying that the WWG&TC was indeed in Banksville. The clubhouse (2007) was now being used as a restaurant, La Cremaillere, and the ancient caddy shack had long ago been converted into a private home.
Before 1940 the name and operation changed to The Westchester Golf Club and men were participants.
edit: My mistake about Bedford. I was remembering the confusion in 2007 when we first came across the course in Banksville. No one knew what it was and the Bedford name came up. Eventually we found out the mistake and that's when I wrote the town.
I wonder if the name "Middle Patent" was used as a nickname to head off any confusion, if, as the historian says, the club changed its name to Westchester Golf Club.