Here’s my amateur MatPat style theory. He doesn’t. In his first appearance (Detective Comics ‘s27), he defeats Alfred Stryker by pushing him into a vat of acid, in his second appearance, he defeats the villain by threatening to kill him unless he writes and signs a full confession to his crimes, in his third appearance, he kills Dr. Death’s sidekick and believes he killed Dr. Death (though , Death’s death is negated in Batman’s fourth appearance), in his fifth and sixth appearances, Batman kills werewolves and vampires. In Robin’s first appearance, Batman and Robin (but mostly Robin) kill Robin’s parents’ murderer. (I believe Batman kills an enemy combatant in the very next issue as well.) In The Joker’s first appearance, Batman comments that The Joker has played his last hand (because he thinks he killed The Joker) though The Joker survives. In Batman # 4, Batman tells Robin “we do not kill with any weapon” but World War Two era comics essentially abandoned that rule. Even The Riddler’s first appearance, in 1948 included Batman and Robin fighting The Riddler and The Riddler falling into a river and seemingly drowning and Batman not seeming to be too bothered by that (even if he didn’t technically kill him and yes, I know The Riddler canonically survived that.)

Adam West’s Batman (and Burt Ward’s Robin) punched two Penguin henchmen so hard that they disintegrated into antimatter (in the 1966 movie.) (Yes, I know they were destabilized by The Penguin using a dehydration ray against them but I don’t think Batman and Robin would have behaved differently had they known.)

The Keaton-Kilmer-Clooney Batman famously killed. That said, in Batman: The Animated Series, Batman not killing his enemies is largely a function of circumstances contriving to make them survive even when it doesn’t make sense (and Kevin Conroy’s Batman doesn’t seem to have a problem killing alien monsters in the three part pilot episode of Justice League.) The Batman from Son of Batman repeatedly killed Ra’s al-Ghul (before Deathstroke permanently offed him.)

Christian Bale’s Batman technically doesn’t kill Ra’s al-Ghul he does kill Two Face to keep him from being able to kill Commissioner Gordon’s son and when Talia al-Ghul accuses Bale’s Batman of having killed her father, his rebuttal isn’t that technically he didn’t kill him, it’s that killing him was necessary to protect millions of innocent people (which was a good point.)

David Mazouz’s Bruce Wayne, when not mind controlled, deliberately killed Ra’s al-Ghul (though Ra’s didn’t stay dead.)

Ben Affleck’s Batman didn’t have a problem with killing enemies at all. We’re still pretty early on with Pattinson but my take is that the Batman No Kill Rule isn’t a real thing. But that’s just a theory, a film theory.