In Batman's case, the role he's taken for himself is most like that of a cop. He catches bad guys, he doesn't judge them. That's the part he takes in cleaning up the mess.
You're saying that because all the other parts of the system don't work, the one that does should be skewed as well. Two wrongs don't make a right, and it's a dangerous foundation upon which to base a society.
Now, if killing Joker is a moral duty, then there's not the slightest reason to hold Batman by that standard instead of the rest of the people who have ever been in situation to kill him.