By going on a murder spree in a downtown metropolis (sorry) and ignoring all reason, advice, or even the origin of suspiciously menacing letters in your mail box, so as to make your very own glowy Spear of Longinus to impale another man with while saying, "Nah, you're a wimp, brah."
Yeah, very inspirational.
No one said that Batman during his low point, before his epiphany, was inspirational. The discussion is about how the story of Bruce's redemption, and in particular the character of Superman, was inspirational. Also, let's get the facts straight. Batman doesn't go on a murder spree in Metropolis. He goes there to acquire the kryptonite being smuggled in at the docks, and he starts to get aggressive to both retrieve the kryptonite and defend himself against their attacks. It's not even made explicit how injured those he goes after are or even if any are actually dead.
The letters Bruce received that were menacing were presented as if they came from Wallace Keefe himself. They said, "You let your family die" with, if I'm not mistaken, un-cashed Wayne Enterprises' checks enclosed. Sure Bruce could have been paranoid and speculated/investigated them as being inauthentic bait, but on the face of it the origin of them would have seemed pretty clear, and the effect would be the same regardless. Bruce feels powerless and guilty about what happened with his employees in Metropolis during the Superman/Zod fight, so that pre-existing turmoil thrown in his face in tandem with Keefe and the entire Capitol being blown up at Superman's hearing, combine to push his buttons.
Lastly, Batman's comments to Superman during their fight are a lot more nuanced than name-calling (wimp). He spends the time verbally and physically attacking Superman for his lack of humanity by forcing him to feel as a human. He makes him fear. He makes him hurt. He does this because he himself has been hurt and has been suffering from PTSD. As such, triggers are a very real thing he has to deal with. That's what "Save Martha" and "You're letting him kill Martha" served as. First, the mention of his mother's name enrages Batman. However, when Lois arrives to provide a human presence -- one that reminds him of his own parents in their last moments -- and clarification that Martha is Superman's mother who needs saving, Batman relents. He realized in that moment that he had become the villain of his childhood trauma and of his nightmares.
Superman's own humanity, humanity that he dismissed as nonexistent before, reminds Batman of his own humanity and heroism, and he makes a promise to Superman to save his mother: his first act of penance. Then, after Superman dies and the world's reaction, he regains faith in himself and the people of Earth, reversing his previous "What falls is fallen" belief to "Men are still good...We betray one another, but we can do better." He starts by not branding Luthor in prison. He continues in JL -- a film where he discusses his regret and remorse and where people like Jim Gordon comment on his improvement -- by encouraging other heroes as heroes and helping them as people (Martha's house, Barry's job). Like Saul of Tarsus who had persecuted Christians, but had an epiphany and began expanding the Christian church as Paul, Bruce is transformed. And it is that change, and the source of that change, that is inspirational.
Superman doesn't show him that any more than any other normal person does. Batman's actions and his turn aren't inspirational, because they aren't dealt with I think. Batman did this stuff, he essentially had a mental breakdown, that doesn't get solved because someone dies. It calls his character and stability into question. That's never addressed and is swept away because the movie wants to pretend like things are all better and not deal with the issue to me.
No one is saying it was solved, just that it was a wake up call. After that moment, Batman is in recovery, which is a process. His efforts in JL are part of that process.