The right answer is that Batman is already an outlaw, but his extralegal behavior is premised on preserving and strengthening what legitimate authority exists in Gotham so that a better and more responsible society can ultimately be built. Batman exists because of the extraordinary circumstances in which Gotham finds itself — a city so tainted by corruption that the local government is incapable of acting in the most rudimentary public interest.
So it isn’t actually Batman’s role to kill his enemies — even those as crazed and sadistic as The Joker. That would undermine the larger project, the restoration of Gotham. Batman’s endgame isn’t a Gotham without crime — it’s a Gotham where the problems can be handled by public institutions rather than costumed vigilantes, where Jim Gordon isn’t an anomaly. That can’t be built on vigilante murder, particularly since doing so would make Batman a target of the very institutions he’s trying to save. The moment Batman decides to kill the joker is the moment Gordon decides he’s going to stop pretending he has no idea what Bruce Wayne really does at night.
To the extent that The Joker is still alive, that’s Gotham’s failure. Batman has captured him time and time again, only for the state to choose the lenience of a stay in Arkham Asylum over trial and state-sanctioned execution. The Joker, after all, isn’t really insane in the sense that he’s not responsible for his actions — it’s not like he kills because he doesn’t know right from wrong. He kills because he thinks it’s funny.
This seems like an overly generous reading of Batman’s objectives with respect to Gotham’s law enforcement. It’s a stretch to say that Batman’s adventures serve to provide Gotham’s legal institutions with more room to function; if anything, the police department is increasingly reliant on Batman, since his tactics invite the presence of the kind of bad guys that are way outta the GCPD’s weight class.
Even if you buy Adam’s argument, it doesn’t seem to me that a police outfit willing to overlook Bruce Wayne’s myriad extralegal, rights-violating exploits — and I think we can all agree that all that high-tech eavesdropping, jumping through people’s skylights and flying experimental fighter planes through the middle of a major city is some undeniably criminal shit — would suddenly see the stealthy elimination of an unstoppable sociopathic mass murderer who cannot be effectively imprisoned as a bridge too far.
I’m certainly not arguing that Batman start murking every supervillain in Gotham (the Riddler and the Penguin, por ejemplo, are more schemers than cold-blooded killers) but the Joker is clearly an outlier. Since there’s no legal avenue that can be taken to neutralize him — he can’t be executed because he’s insane — stopping the Joker requires stepping outside the law, which Batman does all the time, with no repercussions.