You can make the request, but everyone requests the West Wing and while the West Wing is pretty cool, dining in there can be a tad of a letdown.
The biggest problem is that everyone that does not dine in the West Wing visits the West Wing. Sadly, far too many of them care more about taking 638 flash pictures of the room than about your dining experience. The room is also fairly dark, and I have never cared for eating in the dark.
Because if this, I prefer the main ball room. The lighting and high ceilings make it feel much more inviting and I love watching the snow fall that is just outside of the rear window.
If you have to dine in the West Wing, an early lunch is the best time to do it.