Joe, my information shows he built one nine at a time during the period between one in 1922, the other in 1924
I have an article from Met Golfer 1924 stating they were adding the second nine to the 9-hole course operating since 1924
After re-reading the article I think I was wrong about the type membership because among the members mentioned were J P Morgan, the Pratt family members among other big names.
The article refers to the "club" as more informal than its illustrious neighbors.
Here's a good one:
Brookville's property was surrounded by the properties of E D Morgan, H P Whitney, Gen Geo Dyer, (the very impressive) Clarence Mackay and is opposite the Piping Rock Club
I also have a topo map of the course showing clearly the holes and the bunkering