But this is more about Petrobras being used by the government to extract bribes from private contractors, then Petrobras instrumentally bribing/lobbying the government for its own interests.

The Petrolão scandal itself was more instrumental in bringing down Dilma (though ultimately she was not charged for it in her impeachment, but for other things) than Lula.

The charge on which he was arrested (and then dropped) was for a bribe by a construction company (OAS), not Petrobras. Petrobras was at the center of the "Petrolao" scandal, where it took bribes in the form of kickbacks from contractors and passed them on to governing parties.

Looking forward to reading it when it's ready to share.

It also does run ad campaigns on policy issues or its own views on economic and social issues, though they're more indirect and advocacy-adjacent, closer to ESG stuff (but still sometimes timed with policy debates)

Having said that, there are a number of other formal and informal channels through which it can influence policy, since it has direct ties to the government as majority shareholder (e.g. some ministries appoint board members, govt nominates high-level executives, etc.).

Not an expert, but Petrobras is majority state owned, so not legally allowed to engage in any kind of election-related or partisan activities, nor is it allowed to engage in direct lobbying (also, there is no clear legal framework for lobbying in Brazil unlike the US).