In passing, this is an example of where libertarianism breaks down as a practical alternative.
How will you determine what these costs are? It might be simple if I have a fishing business in Louisiana, but what if I'm a bar owner who depends on the dollars of hard-drinking oil workers when they're on shore? What if I'm just depressed by the whole thing and miss work? There are millions of people living in that area, and the gulf coast economy is about 2-2.5 trillion. You simply can't handle every individual case in court, it's going to be a series of class actions. And without some kind of judicial standards for how those should be handled, the result will be arbitrary at best.
Additionally, BP will argue that oil drilling is fundamentally risky and the residents of the area liked it when the going was good. So there will be further arguments about the degree to which it is really their fault. Again, without agreed standards in place the outcome of such wrangling will be arbitrary.
Finally, whatever a court orders BP to hand over as compensation for its poor drilling practices is purely hypothetical unless there is some way of enforcing it; BP might just sink the remains the of the rig, announce the failure of its 'gulf operations subsidiary' and just turn its attentions to operations elsewhere, where they have never had an accident. Perhaps they'll just blame the whole thing on inferior oil workers in that region.
So without standards for judging claims and liability, or mechanisms for enforcement of judgments, the situation will not be properly resolved. Those standards and mechanisms are the basis of laws, and we elect governments to serve as our agents in establishing and promulgating them.