Private Room for Lunch Near Santa Anita Racetrack
There is a private room at Full House Seafood Restaurant in the 99 Ranch Plaza on Duarte and Golden West. It is a dim sum place, but I don't know if you can reserve the private room at lunch. It can be a little harder to flag down the carts if you do get the room. Another Chinese (non dim sum) option is Chang's Garden. They have a separate semi private room. By semi private I mean that there are two tables in their separate room so if your party only takes up one table, there may be another party next to you!
Did you try the Group Events department at the track? I used to work there for 5 years and they should be able to accomodate you somewhere, although it may not be totally "private". They do have minimums for renting out the Club Plaza for instance.
Are you holding a meeting while at the races? May not be the best place to hold a business meeting, but a lot of fun for a social event.
You can also ask the GE department for recs in the area, if they can't give you a private meeting space. It's been 6 years since I lived right there, so my restaurant recs may not be current. Also, try the mall, they have some new restaurant anchors, and may be able to accomodate your group.
Good luck at the races!