There is no 'correct.' You are allowed your opinion, I am allowed mine.
Not if the moral relativism used to examine your competing opinions is only Descriptive Moral Relativism; meaning, you and your debating opponent hold different positions of what is right (ethical) and what is wrong (unethical) behavior, but while you are allowed to form your own opinion in this matter, one of you could still be right and the other in error, in a universal sense of morality.
The problem I have with the Meta - something (I don't feel like looking for an old philosophy textbook to find the exact term right now.) Moral Relativism, is that it eliminates the need to ever question or reexamine one's beliefs, leading to a limited possibility of growth or greater understanding for either the individual or the society clinging to its own set of rules and beliefs for the sole reason that they are
their beliefs, not because they are better, or more effective, or true in a universal sense.
Now that I got that off my chest, I wonder if the Winter Knights mantle loosened up Harry's inhibitions to kill something that was not presenting an immediate threat to his life, made it a little easier to shoot first and ask questions later.
Even if that is not the case, I don't think Harry actions were as cold blooded or evil as the OP suggested. Harry had just survived what should have been a fight to the death, and a young women whom he liked; even if he didn't know much about her, had just had her life threatened by one of the Sidhe who had a buddy who was more than willing to do the job. Also, we have never seen any of the Winter Court show more than cursory respect for mortal life, and Harry was standing at the seat of their power, so why shouldn't he have reacted with extreme prejudice to a provocation, even a verbal one? Anything less would likely have likely led to worse situation for him.