There's a universe where Batman was basically the punisher and goes around killing people.

The Grim Knight kills all the corrupt cops, mob bosses and villains. He also blows up blackgate and Arkham. Gotham ends up super safe, but somehow more of a dystopia than regular Gotham. Everyone lives in fear, and Batman has a 1984 setup where he spies in everyone and uses Waynetech to kill people.

Commissioner Gordon ends up arresting him with the help of info given to him by Alfred.

Would it make sense for someone to kill the Joker? Yes. However Batman is, in addition to being an anally retentive genius perfectionist, somewhat emotionally unstable. Dude isn't the type of person who'd be able to kill the Joker and call it a day.

That being said I'm pretty sure the in universe reason right now is that killing the joker releases a gas that turns whoever is near it into another joker. If Batman killed the Joker, the gas would cause him to turn into a villain who can singlehandedly kill every other hero on earth. He would then continue to pull bullshit out of his ass until he gains the powers of Doctor Manhattan and almost destroys all of existence.