I don't think Lane had anything to do with that, but Luthor was directly involved in the Brainiac attack.
Luthor was working for Lane. The deal was that Luthor works for Lane pretending to work with Brainiac and in return he gets pardoned and gets control of LexCorp.
Uh, I'm confused. Is the Pentagon all of a sudden not part of the US government now? Giving him a blank check is the exact definition of approving whatever he's going to do with it! They are directly culpable for his actions whether they knew what he was going to do specifically or not. In fact, off the top of my head I can't remember a single instance where it was ever mentioned that Lane was acting against the wishes of the government.
Even in real life, the Pentagon has black operations that they don't have to explain and don't obtain full approval by Congress. The government gave the Pentagon the ability to combat the potential Kryptonian threat, but they didn't exactly know all the details of what General Lane developed such as creating a suicide bomb within Reactron.
And saying that all of this falls solely on the shoulders of two individuals is incredibly disingenuous. Lane didn't singlehandedly destroy a planet, he had the support of hundreds of military personnel and scientists planning genocide with him every step of the way. Not to go all Godwin on everyone, but blaming all of this on Lane and Luthor is the equivalent of saying that Hitler gets all the blame for everything the Nazis did.
Well the entire Project 7734 does deserve blame, but the major figures of the group were Sam Lane and Lex Luthor (the same way we blame Hitler for everything the Nazis did). Project 7734 went behind the US Government's back under the orders of Sam Lane, who is the equivalent of a renagade general.
To say that Zod was justified in taking over Earth over the actions of a rogue group that took their mandate too far without proper approval (the US Government did not authorize them to commit genocide or provoke a war, just to protect Earth) is absurd.
I can't imagine Superman allowing public opinion stopping him from doing the right thing.
Also take into account that he just fought the last surviors of his people who were just as strong as he was, his race has now been doomed to extinction, and he hasn't seen his friends and family in over a year, Superman really wasn't in the mental or physical state to go after Luthor either.
The best way for Superman to get Luthor to pay for his crimes is to expose what he and General Lane did that provoked the 100 Minute War, which was what the end of the New Krypton Saga said they were going to do (and never followed up on).