In general, no, but Battlefield Bad Company 2 is known to like alot of CPU power, and extra cores as well. The game likes quad core CPUs. There's a thread here somewhere where someone went from a Core 2 Quad to a Sandy Bridge and saw very large gains in this game. A little bit of struggling with a Core 2 Duo will be normal.
Overclocking it should help. You might not double your maximum, but your minimum and average might smooth out a bit (and those two things are more important than maximum). From the percentage numbers given, the CPU is the bottleneck, but not to a large extent.