Does Lex Luthor know Batman's identity?

Question

It's clear from the events of Dawn of Justice that Luthor knows who Superman is, given that he eventually

Kidnaps Martha Kent as leverage,

but there are times in the movie when he also seems to know who Bruce Wayne is. For example, when Clark Kent and Wayne meet for the first time at one of Luthor's parties, Luthor acts as if the meeting is somehow historic. At the end of the movie, when Batman confronts Luthor directly, Luthor responds:

LUTHOR: This is how it all caves in. Civilization on the wane; manners out the window.

The way he puts emphasis on the words, and where he pauses, it sounds like he's saying "caves in Wayne Manor", which would be a direct reference to the Batcave. But there's no real evidence by his behavior that he knows Batman. It's possible he just doesn't care yet, but it seems like Batman got the best of the situation

by sending Luthor to Arkham

and Luthor didn't resort to threats of blackmail to avoid it.

Is the intent that Luthor knows the identity of the entire Justice League (given that he also had access to the other metahuman files at one point), or was this just overly-clever writing?

Answer 1

At one point in the film (during the meeting with Superman on top of LexCorp) Lex says something like

... but it did not take much to push him [Batman] over actually, little red notes, big bang, "you let your family die!"

And in another scene (during the hearing in Capitol) we see Bruce Wayne receive a returned check from the guy he saved from Wayne Tower, saying

You let your family die! (written in red)

In the first scene Lex is referring to Batman, in the second - Bruce Wayne receives the note. I think it's proof enough that Lex knows everything about everyone.