You mean the obvious threat of Luthor stealing the Kryptonite, blowing up the Senat and keeping tabs on metahumans? My bad, you're right, he should have gone after the guy in the cape who keeps flying around and saving people.
Sorry, but the character "they created for this movie" isn't Batman. Batman, in almost every iteration in media has never been portrayed as this outright murderous, non-logical, lunatic.
Even Burton's Batman wasn't this bad.
Burton's Batman was much more of a direct killer. This Batman, while he certainly does not mind casualties as a consequence, in the course of action, but he's FAR from the Punisher, like some have said. The Punisher goes out of his way to kill badguys. He's never leaving people for the police.
This Batman, at least after he's committed to destroying the Superman (we don't see him cause any casualties before this) doesn't go out of his way to kill, he just doesn't seem to care if it happens in the course of achieving his ultimate goal of preventing Superman's inevitable transformation from the Knightmare. Well, with the exception of the completely pointless and unnecessary shot of him gunning down the guys outside the warehouse.
Again, I think they absolutely could have told the story without it, and I DO object to/disagree with it as a creative decision, but, despite causing a significant amount of death, he's not the mindless psychopath so many are attempting to portray him as.
As for Luthor, when are all those things made clear to Bruce, to be his fault?
Bruce does NOT know that Lex blew up the senate, he believed, as everyone did, that it was entirely the actions of Wallace.
All Bruce knows of Luthor is that he's tracking, and concerned about, meta humans, EXACTLY the same as Bruce.
While he may not TRUST Luthor with the weaponry he may be developing to "defend" against the Kryptonian's, he hasn't seen anything of Luthor's goals (outside of Luthor's desire for power) that indicate he's attempting anything different than himself.
So why WOULD he turn his focus to taking out Luthor, whom he's only seen to be trying to do, essentially, the same thing Bruce is?
He doesn't know that Luthor has ALSO been creating international incidents, and suicide bombers to further his cause, and taint Superman's image.
This criticism makes no sense.