It seem like you have the fancy places covered, so I want to suggest some of the others.
Columbus Park is not as fancy as some of the top end places mentioned on this thread, but it is a nice atmosphere and the food in great.
Another nice option in my opinion is Brasitas. The have a private room upstairs. Brasitas (Spanish / Latin food) is more of a local place, but great food and sangria. It has gotten many great NY times reviews. Upstairs is nice and separate from the rest of the restaurant.
Other places with Private Rooms are Ferrante and Mitchell’s Fish Market.
Good luck with it