Small Private Room on the Westside
Hop Li Chinese restaurant on Santa Monica Blvd has 2 private rooms. They have a minimum, but it should be no problem to keep it affordable for your group.
The Chinese restaurant at the NWC of Barrington and Wilshire has a private room with 2 tables for 10 each. Have not been there in it's current incarnation.
Vinosnob mentioned Enoteca Drago in Beverly Hills. Wine can get pricey. They offered prix fixe $32 menus at one time. On a non weekend, they might allow your group to go with some kind of Prix Fixe menu.
Less pricey Italian, with more chain restaurant food but further away in Santa Monica is il Fornaio and Buca di Bepo.
410 N. Canon Dr., Beverly Hills, CA 90210
You don't mention how much is not too expensive or the ambience/food you're looking for.
Westside Tavern is a good suggestion. Musha has a small tatami room in the back that can be reserved for a party of that size.
Couple others on the westside that might work - Upstairs 2, Upper West, Canal Club.
1725 W Carson St, Torrance, CA 90501
2311 Cotner Ave, Los Angeles, CA 90064
10850 W Pico Blvd, Los Angeles, CA
3321 Pico Blvd, Santa Monica, CA 90404