The issue with this is when superheroes refuse to kill, then others die to pay for it. Batman won't kill Joker. But Joker will escape again, and will kill again. So by refusing to kill, Batman is enabling Joker to kill.
I think you're letting the people who failed to keep Joker incarcerated off the hook, but a bigger issue, at least for me, is that you're making Batman the arbiter of who should die. Even if Joker 100% deserves it, we only know that because we've witnessed his crimes from artificially omniscient point of view. But then who else should Batman kill? If we say he should kill the Joker, we're saying he should kill whomever he thinks deserves it. So either we're trusting him to be infallible in his evaluations of who should die, or we're willing to accept the innocent casualties (and if it's the latter, then we should be willing to let Batman take the Joker alive in the first place). We'd be trusting him to not only determine who's committed a crime, but which crimes deserve death and which criminals are recidivists and/or irredeemable (and, come to think of it, if we're trusting Batman to determine who's irredeemable, then we're also trusting that Gotham's rehabilitation mechanisms are infallible - and we're back to letting whoever was holding The Joker in prison off the hook - there's a circle of logic here that just doesn't compute).
Not incidentally, you're also asking Batman to take on a great emotional and psychological burden. Killing someone is a strong risk factor for developing PTSD. That's one reason committing "suicide by cop" is a crappy thing to do (in addition to, y'know, committing suicide). And what does it do to him if he does kill an innocent person by mistake? What if Batman kills some of Joker's henchmen and then later we learn that Joker was holding their children hostage? If you've seen
Daredevil, you may remember the scene where Wesley shows the prison guard the live video of his daughter. Fisk was maneuvering two of his victims against each other, in forcing the prison guard to kill Karen. The assassin who killed himself because he blurted out Fisk's name probably did that because Fisk knows where the guy's [wife/father/niece/baby] lives. Like Kaizer Sozhe, Fisk won't merely kill
you if you screw him.
As well, Batman sits at the head of the table of the "Batman Family." If we're giving him license to kill, what precedent does that set for Nightwing, Batwoman, and Robin? iirc, Cassandra Cain/Batgirl, The Red Hood, The Huntress and Azrael are already Bat-family vigilantes with impulse control problems that Batman has had to shove back in line. If he starts killing, all of them are off the leash.
Finally, if it becomes known that Batman and his cohort kill, it won't be long before Ra's al Ghul or Hugo Strange or one of the other especially cunning villains deliberately tricks one of them into killing someone for them.
And then Clark, Diana, Oliver and Barry show up, because Bruce & Co. have become the villains (see: season 3 of
Jessica Jones).
This shouldn't have been a deleted scene.
