Need small private room for VIP in midtown
There aren't that many tiny private dining rooms so you may need to expand your geographical boundaries. Wine room at Maialino or Skybox at Daniel might work if you can go downtown or uptown.
2 Lexington Avenue, New York, NY 10010
I don't think Daniel open for lunch.
You might want to look into Eleven Madison Park. The food and beverage minimum for lunch in the smaller private room is quite reasonable compared to other three or four star restaurants.
Eleven Madison Park
11 Madison Ave., New York, NY 10010