Thanks for the input, everybody! We wound up at the Berkshire Room, which turned out to be perfect -- they have some semi-private spaces off to the side of the main bar area, and our group wound up in one of these.
The Continuous Negroni is delicious, a couple of folks in our party got drinks they really liked by opting for Dealer's Choice, and my husband (still getting used to saying that) loved his Old Money (barrel-aged, bourbon, aperol, walnut liqueur, and allspice). We were there from around 9:30–midnight, and while the place was buzzing it wasn't insufferably sceney -- just fun. Thanks to daveandrews3 for setting up the reserved space for us. I think we'll all be back.
The Berkshire Room in the Acme Hotel
15 E Ohio St
(between Wabash Ave & State St)
312-894-0800