In regard to Batman's arc and the resolution of the fight:
When we meet this Bruce, he's a jaded and angry shell of his past self with a narrow-minded view of the world around him due to past experiences and, most recently, the Black Zero event in Metropolis for which he was helpless in saving his employees. He feels that his crusade hasn't made a difference and his view on Superman is seemingly a manifestation of that fact that he's essentially lost his faith in humanity, in many ways.
20 years in Gotham
weve seen what promises are worth. How many good guys are left? How many stayed that way?
This hopeless mindset has led him to become more callous, brutal, and even cruel in his methods. He's clearly not the same kind of man or hero that he once was. Alfred says something along the lines of "this is not how things always were" and, of course, says this:
"That's how it starts. The fever, the rage, the feeling of powerlessness that turns good men... cruel."
While it may seem that he just has this quick change of heart about Superman, there is more to it than that and also a good reason for it. Batman experiences what's called a moment of clarity -- a sudden and deep acceptance of a truth that had been impossible for him to see up to that point.
When he hears the name "Martha" and learns it is the name of Superman's mother who is in danger, he not only suddenly recognizes the humanity within Superman, but also realizes that it was HE who had lost the humanity within him. He had become exactly the kind of monster he thought he was fighting against. He lost the fight with Superman before it even began, partially because the rage and emptiness inside him clouded his judgement to the point that he allowed himself to be so easily manipulated by Luthor -- something I'm guessing wouldn't have happened in the past -- and failed to recognized or believe in the potential for Superman to be inherently good.
This moment -- along with what comes after (fighting side by side with Superman and watching him essentially sacrifice himself) -- does not only change his opinion about Superman, but it seemingly restores his faith in humanity and gives him hope again, as evidenced by the "men are still good" conversation he has with Wonder Woman. Beyond that, this experience is what drives him to want to seek out the other metahumans to join him and Wonder Woman in protecting humanity -- rather than continuing on working alone in near isolation. Let's remember that after the various personal losses he had suffered over the years (including the death his partner Robin and most likely people close to him who turned bad or betrayed him), it's a huge reversal for him to able to trust in other people enough to want to work and fight side by side with them as Batman.
Lastly, in his final scene with Luthor in his cell, it's very telling that he chose NOT to brand Luthor's skin with his symbol (which was said the be a "death sentence" for criminals in prison). To me, this indicates that he's likely leaving behind the "executioner" aspect of his methods and likely going back to being the less brutal, more hopeful hero he once was.
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