The real question is "Will you take the law into your own hands?"
That question, I dunno. In a purely theoretical discussion, I would like to say yes. This subject is one that I personally find frustrating about the Batman universe. Corrupt legal system or not, how many bodies does one person have to pile up before the death penalty is considered appropriate? Crazy or not, doesn't matter to me. At some point, (personally when someone has wracked up a body count in the double digits or more), their sanity shouldn't be an issue. They are a clear and present danger to society, and given the revolving door system of the Mental Health Industry (not counting breaking out in comic book universe), I think putting them down is the overall better choice. The likelihood of them getting out is ridiculously high in a comic universe, and they will, as you say, kill again and again. At what point do you say "the value of this one crazy murderer's life is greater than the mountain of bodies he has made" ? When does that scale tip the other way?
I don't know, personally, I would like to think I'd kill him. Assuming all the variables you mentioned:
1. Well known, documented history of mass murder
2. Inability of justice system to properly protect the public by keeping him locked away.
3. Certainty that he will continue to kill again. (Very likely including myself if he gets out of this Perfect Situation)
I would say yes, I would kill him.
I do find it funny though, that the regular police force can't seem to shoot him. I mean either they are Stormtrooper accurate, or prohibited from using lethal force. If they are prohibited from using lethal force, then why do they have guns?