La Figaro fits the bill. IIRC a lot of their business in large parties. Maybe Bass St. Steak and Chop in Moline? I believe they've got a private room. If not the booths are very tall and tables get a lot of privacy.
EDIT: totally missed aquapacat's post on La Figaro, but I did think they did parties. You might want to call and ask. They've got great food there.
I'm glad I stumbled on to this post as I grew up there (but haven't lived there in 25 years), and the few times I fly back, it would be nice to know what are the good digs there.
I DO remember a place called The Belgium Village in Moline, that had by far, the best Reuben sandwiches I've ever had. Homemade bread of all kinds, and absolutely ridiculously large Reubens that came in several sizes and combinations.
Last time I ate there was 8 years ago, soooooo. . . .?
Sorry to be late in replying.... Trattoria Tiramisu has a room - it's in Bettendorf. I seem to recall that Le Figaro doesn't do any special dinners. You could rent out SereniTea in Davenport - they do a lot of private parties, for a daytime event on a Saturday. Other places worth checking are Cafe Indigo in Davenport, Red Crowe Grille in Bettendorf, Ganson's in Rock Island, and Cafe' Fresh in Moline.
1804 State St, Bettendorf, IA 52722
4925 Utica Ridge Rd, Davenport, IA 52807