In asking this question, you have added a layer of interpretation onto the text that I do not see there. You introduce the word "want" which does not exist in the text.
In fact I don't see any indication in the text that Paul has a desire for personal vengeance or that he wants Alaxander to get what's coming to him. In fact is is quite likely based on what we know of Paul's responses to other people and his consistent teachings on related subjects that he would actually want the opposite for him.
However Paul is under a charge and has a responsibility here.
He must warn the congregation about a real danger to its health. We are sent out as lambs among wolves, but we are also told to be wise as serpents. That includes identifying the wolves. By name if need be. As a shepherd Paul had to guard the congregation against somebody who would damage it.
He must also warn Alexander that the consequences of his actions are eternal. People must be informed that while they may not experience punishment in this live, that does not excuse their actions and they stand under God's judgement. In phrasing his warning this way, he also makes it clear that it is not he (Paul) who will seek or meter out justice while still making it clear that justice will be served.
This seems consistent with everything else we know from Scripture -- that vengeance is God's and that those who oppose his work on this earth will find themselves opposed by God. Paul can at once long for ultimate justice to be done and God's people set free from their persecutions and even rejoice in the knowledge that the wrong will be set right -- while not having a personal desire to see somebody punished and not raising a finger against them.
Protecting the church and warning those in the world that they will be judged by God is not the same thing as what you imply. There is nothing here to indicate that Paul "hates a man" or "wants him to suffer" and his response is not outside the bounds of Christian Love.
Particularly in the West, we often compromise truth in the name of something we call "love", yet it is no love that does not proclaim the truth including making clear the severity of God's anger against those who oppose him.