Use the freedoms that liberal democracy and capitalism give us to fight them back. Expose their corruption, counter-lobby our representatives, use the courts where laws are broken, vote. There is no magic bullet, corruption will always have to be fought. The question is which system gives us the best framework to fight it in. I say a system that respects individual freedoms.
I don't at all want to give the impression that I'm against liberal democracy or individual freedoms. I'm with you there. I'm just not convinced that they're as compatible with absolute private ownership of capital as they are billed to be. And I don't by any means think nationalization is some silver bullet or even the ideal strategy - just trying to think it through.
But setting that aside for the moment, it does seem that up to now, exposing corruption, counter-lobbying, and courts have been our primary tools for fighting back against the likes of Exxon. How is it going so far? It does not appear to have worked in time to avoid major catastrophe. Is there a limit to their effect? And when we talk about counter-lobbying and using courts, what outcome do we hope to achieve with a company like Exxon? Do we want them to pay for the externalities of climate change?
The reason we haven't solved climate change is just because it's an incredibly hard problem, and we just flat out don't have a political consensus and clear mandate from the people to take the sort of drastic and incredibly costly steps it would take. Changing direction like that is a political problem first and an economic problem second.
As for Exxon, they may be scumbags and need reigning in, but basically they are just a business providing a service. We all consume fossil fuels all the time, and we need someone to buy them from. The way to reduce fossil fuel dependency isn't to just kill the oil companies, we'd kill our economies with them. We need to wean ourselves off the poison. We need to stop being their customer, not blame the shop for selling us the goods we went to the shop to buy.
I'm not sure what you mean by absolute private ownership of capital. We don't have that in an absolute sense at all. We confiscate capital through taxation (which I'm fine with by the way), we regulate markets (which is a necessary function of government), the government controls the money supply and directly manipulates financial markets all the time as is needed. The government is a massive employer and huge spender in the economy. I'm a Brit actually, but all of that is true of every liberal democracy. Heck, I even support the NHS, I think it's far better than the US model. However capitalism is fundamentally about individual rights and freedoms, not absolute ones, but valuable ones. I'm a wishy-washy centrist, and proud of it.