Firstly, it should be noted that virtually any vision you have for Batman with respect to killing, at SOME point the comics have gone there. In his earliest days, Batman (and Superman) killed all the time. However, relatively quickly a newer standard was set of Batman not killing, ever, and in fact often doing everything in his power to save the lives of criminals (although, to be fair, a lot of the time he still did ridiculously violent actions that could easily lead to death, and where it was only by editorial fiat that nobody he beat up ever had a fatal concussion from it).
There are a few exceptions, and almost always, they were overturned. One notable case is the KGBeast, who, in Batman #420, the writer intended that Batman lock him in a room with the intention of letting him starve to death.
As this article describes, the next writer seemed to reverse it and retconned it and had Batman say he informed the police hours later.
Now, where did Zack Snyder get his justification for Batman killing? Well, in this article, you can read the following quotes from him:
I tried to do it in a technical way. There’s a great YouTube video
that shows all the kills in the Christopher Nolan movies even though
we would perceive them as movies where he doesn’t kill anyone. I think
there’s 42 potential kills that Batman does! Also, it goes back and
includes even the Tim Burton Batman movies where this reputation as a
guy that doesn’t kill comes from.
It should be noted that the reputation that Batman doesn't kill does not in any way come from the Tim Burton movies (where often the same cavalier attitude towards killing is present).
So, I tried to do it by proxy. Shoot the car they’re in, the car blows
up or the grenade would go off in the guy’s hand, or when he shoots
the tank and the guy pretty much lights the tank [himself]. I perceive
it as him not killing directly, but if the bad guy’s are associated
with a thing that happens to blow up, he would say that that’s not
really my problem.
It should be noted that many of these things are still considered murder.
A little more like manslaughter than murder, although I would say that
in the Frank Miller comic book that I reference, he kills all the
time. There’s a scene from the graphic novel where he busts through a
wall, takes the guy’s machine gun… I took that little vignette from a
scene in The Dark Knight Returns, and at the end of that, he shoots
the guy right between the eyes with the machine gun. One shot.
It should be noted that the scene in The Dark Knight Returns does not happen as he remembers it...
It certainly looks bad, but there's a detailed analysis of the scene here, but in very brief terms:
No, in the scene in question there was no on-panel killshot, there's evidence from other scenes that the thug was not killed (though was probably shot), and in fact there's later a major plot point in the series that hinges on Batman still not being willing to kill, subduing a threat is enough. If Batman was the kill-happy vigilante Snyder thought he was, the scene would make no sense at all.
So, basically, Batmans lack of morals in Dawn of Justice seems to boil down to Zack Snyder not understanding the source material (given his treatment of Superman, "choosing to disregard the source material" is also a potential option, and I don't even argue that it's necessarily a bad thing to disregard source material, but his specific justifications suggest a lack of understanding is the culprit here).