The definition of a vigilante is that he IS working outside the law. Therefore he is not bound by due procedure the way law enforcement is. The Bat consistently tries to have it both way and FAILS. He has to endure being reviled as someone who makes a mockery of the law while privately binding himself within the law. You just can't have it both ways. Either you can get your hands dirty or you can't.
I maintain that killing someone like the Joker or Zsas wouldn't be murder, it would be a "putting down".
If Batman let Joker take the fall for it, there would've been a criminal going unpunished and free to commit other crimes. The Joker was deliberately set up by one of his previous victims in a manner that wouldn't allow him to be shieled by reason of being legally incompetent any more.
Did I miss the moment where Bruce acquired Arkham Asylum somehow and failed to strengthen their security measure?
You missed where the Bats all keep taking insane criminals who are also tlaented escape artists to a place which consistently embodies the failure of the system to contain these criminals in any effective way. If you decide to work above the system because its broken, then you're supposed to go the whole hog and make sure they can't escape either. That Bats only takes on the role of a cop, when he has the resources to do much more than that, feels like a pretty half-assed way of trying to change the system.
I'm currently fleshing out a plot bunny where Tim has become Batman and, tired of seeing the top-tier villains escape and cause havoc time and time again, uses Waynetech to build his own ultra-secure prison for criminals who have been deemed too dangerous for either Arkham or the Slab to contain. He has a secret contract with the governments of the world to contain them. You can't tell me Bruce couldn't have done the same, if he were willing to bend the rules a bit more.