A Battery's C rating is only an indicator of internal resistance. If you have a battery with a high C rating, it will be more capable of supplying large current, but that current is dictated by the ESC and motor. The battery only gives what is demanded of it. The only time a high C rating is necessary is for racing in classes like 4wd SC, mod TC, etc where current draw is huge. Some Spec classes draw large currents too. So a large C rating will help, but the only thing it hurts is the pocket book. If your C rating is low, like 20C, you will just be slower; so long as the basic set-up rules mentioned above have been met.

On another note, you said your car came with a 7.2V 2000 mAh battery. That is a NiMH battery. The batteries that are C rated are Li-Poly. Do you know if your ESC is Li-Poly compatible, i.e. does it have Low voltage cutoff settings? Without this crucial function I would not run Li-Poly.