The heart of my argument is that killing killers is (1) illogical,
If they're trying to kill you at the time, it's very logical indeed. If the damage has been done, and society's doing the killing, you may be right - but to have standing to complain about killing murderers being illogical, you should point out what treatment of them is logical. Giving them 40 years free room and board doesn't really cut it either.
and (2) doesn't deter murderers,
It deters that murderer very effectively indeed. You appear to be right that it doesn't deter the others very well.
and (3) contributes to the societal attitude that people are disposable, and thus that killing isn't really all that bad.
A society that fights wars already has that attitude so ground in that it doesn't really matter. A society that doesn't fight wars becomes extinct in a generation or so. It's sad, but I see no other choices.