Maybe part of this comes down to how you view the Batman mythos before going into this movie. I feel like my take on Batman as a whole matches up with Snyder's and also with how Snyder wanted to portray him here. Batman is intelligent in certain ways, yes, but he's very lacking in others. His parents murder messing him up has always been one of the biggest parts of his motivation and narrative, and this movie pushed that a lot, too. We've had decades of GET OVER IT BATMAN sort of memes. It's odd that we'd expect him to here suddenly. It's a big deal. It's one of the biggest deals in his entire character and the mythos of Batman.
Now, I don't subscribe to the thought that Batman didn't entirely think of Superman as not a human. There are a few other factors going on. The big thing was that when Wayne tower was falling in Metropolis he what sort of person was he there helping? A little girl who lost her mom. An orphan now, because of Superman. That's basically Batman's trigger. It's something he's very much not rational about and never has been in almost any story I can remember. Yes its a bit troubling, but it always has been. Thats the cornerstone of his character.
There's a rather important conversation with Alfred that talks about his other real motivation. He talks about how Superman is good now, but then says "How many of the good ones stayed good?" reflecting on all the stories we know about Gotham and Batman. This guy has been going after thugs for decades. He barely sees the good in anyone anymore. Everyone has either died or turned bad. So it isn't just that he's not "human," it's that he can't stay good. No one stays good. Even Batman himself doesn't stay good, as evidenced in this movie. If Superman is creating orphans and destruction like that while "good," what could he do if he ever turned? A lot. More than all of Gotham's thugs combined. It could do so much that Bruce is now worried about his legacy. Nothing he's done matters up until this point once you have gods that can fly through buildings and practically blow up the planet. It's full on existential crisis at this point. Thats why his behavior gets even more erratic and violent. What does it matter if hes truly just if someone like Superman exists? So he starts branding people and his no-kill policy obviously gets a bit flimsier. It has to get flimsy. He cant lock up superman. In order to rid himself of this, and in order to have that legacy he spoke of in the film, he has to KILL Superman. Once he sets out on that quest and once that crisis begins there isnt any turning back.
That crisis also builds on another big part of the Batman mythos that I subscribe to. He's usually portrayed as only barely more sane than the people he's locking up. He's a nutcase. He dresses in a bat suit and scares criminals based on his parents being killed as a kid. His motivation belongs in the same psych files as those he goes after. This movie touched on him actually acknowledging that fact. He even states it when he says, "We're criminals, Alfred." So to me the Batman mythos has largely been playing with this mirror of Batman and his "enemies." Often it plays with questions of "What is the difference between them," "What is the same about them," and also "Does Batman see any of these things himself?" I loved that mirror effect when it's pointed at Superman.
That mirror is partially whats happening when Martha is uttered. As I said up there, orphans and the death of his parents are kind of the biggest deals to this man. Its the first event we see when the movie opens. Its why hes Batman. Hes not entirely rational about it. It isnt that martha is just uttered that he stops, its when he hears that its Supermans mother. Superman is no longer a faceless god. He has a mother. I know youre laughing now because obviously hed have a mother! Well I mean he is an alien so really who knows
. In all seriousness though, it holds that mirror up to Batman. It shows him that they are the same. It forces that to his view. Not only that, though, it gives him a chance at redemption. His entire being is built on the helplessness that he couldnt save his mother and father and now he has a chance to save Martha. Its redemption for him. Thats why its so important that he has to be the one to save her. He doesnt just let Superman go at this point to save his own mother. HE has to do it.
And then I just have to say I love how the arc closes out, then. Superman, by sacrificing himself, proved Batman wrong and showed theres still good in the world. Superman didnt turn bad. Despite the world being against him at times and despite being hunted by Batman, Superman stayed on the righteous path and ended justly. He became the beacon of hope that his symbol was supposed to be, not only to humanity but to Batman. Batmans path is now righted. The movie bookends itself with two funerals that have a profound impact on Batmans path to justice. In the beginning its the death of his parents and then in the end its Superman. With the first he took the path to becoming Batman because of his failings and with the second one he reclaimed that path again because of those failings. As he said in the end, "I failed him in life. I won't fail him in death."
In the end he has hope from Superman and he's redeemed because he saved Martha. This is perfectly shown by the bookend quotes in the movie. It starts with "There was a time above, a time before. There were perfect things, diamond absolutes. But things fall. And what falls, is fallen." and then it ends with "Man is still good. We break things, tear them down, but we can rebuild. We can be better, we have to be." Those quotes show the stark difference in his mentality and in his path.
The other cool thing about this path is that Batman is a metaphor for the people of earth throughout this, too. He represents mankind, and so by Superman being hope for Batman, he is hope for humanity, too. Throughout the movie its mentioned that Batman represents humanity (Lex jokes about this). I think thats largely what Snyder was trying to get at. Batman is one of the most human superheroes that exists so I think hes a perfect metaphor. Like humanity he wasnt ready for gods. He wasnt ready for Superman.