I think I found my bottleneck. 1GB VRAM. I bought the bargain basement model ($141 on sale) and BF3 is using 985GB.
According to this, Battlefield can want more than 1GB on Ultra: http://www.gamepur.com/news/5768-dice-battlefield-3-utilizes-more-igb-vram.html
That said, I don't regret saving $50 to lose a few fps. I just played a few matches at 1680x1050 with everything on the highest setting except MSAA at 2x and it played perfectly. So this thing is a beast for $140.
Also, the one site that benchmarked this exact card was getting better results than me, but I just looked closer and they had used a i7-3960X, so that's just a tiny bit different....
I'm at least getting their stock results by overclocking my card, so it's not all bad.