"Big oil" is not a person. These companies are made up of people, who are motivated differently than the entity (entities) as a whole would be.
Specifically, any change in public sentiment would be rather small and only accrue over the long term. It would therefore be of little use to, say, some PR executive at Shell.
"Big oil" is also made up of a number of companies. Any benefits would be spread out among all of them, while any costs would fall to just the one doing it.
As for costs: the big one would be the damage to your reputation, as an individual, company, and industry, that would follow if the public found out. I'm having trouble imaging a scenario where you run such an operation, certainly involving a dozen people or so, without ever pissing one of them off sufficiently to see them calling a reporter.
I don't know what's going on here. The tobacco industry comes to mind as one that engaged in such tactics even though a lot of the above was true for them as well. But "big oil" seems more diverse, including foreign firms and governments, companies that embrace the future such as Shell as well as more "conservative" donate-to-republicans-and-hope-this-will-blow-over kingdoms of darkness such as Exxon. Tobacco was a far more like-minded, sometimes family-owned community that would have been easier to co-ordinate in such conspiracies.