Do you know if the Oak Room has a bar menu?
Technically speaking, the Oak Bar serves a "lighter fare" subset of the Oak Room menu, which means that burgers should only be available at lunch. In practice, they will serve you pretty much anything off the Oak Room's breakfast, lunch, or dinner menu in the Bar if you ask. It's not the best $15 burger in town (they use "Kobe"), but it's pretty good, and the atmosphere in the Bar is unique.
Oak Room menus here: http://www.theoakroom.com/?l=0,1,2,19
You also might want to note their evening dress code: "No shorts, hats, cut-out/tank-top t-shirts, or sport sandals."
I don't think they do. Someone please correct me if I'm wrong. I've found that whenever I've indicated I would like to eat at the bar they have a tendancy to gently yet firmly usher you towards the dining room. Food always great so I really don't mind. I go to the bar after for a moscow mule.