The passage you quote is very near the start of a new topic in Paul's letter. 2nd Corinthians 8-9 covers the topic of giving; particularly giving to the church in Jerusalem. The context leading into the passage in question in 2nd Corinthians 10:1-2 (ESV):
I, Paul, myself entreat you, by the meekness and gentleness of Christ—I who am humble when face to face with you, but bold toward you when I am away!—I beg of you that when I am present I may not have to show boldness with such confidence as I count on showing against some who suspect us of walking according to the flesh.
So the topic Paul is directly addressing is the perception among the Corinthians that he is walking according to the flesh because of the way he writes his letters to them (i.e., boldly). This thread started back in 2nd Corinthians 7:8-9 (ESV):
For even if I made you grieve with my letter, I do not regret it—though I did regret it, for I see that that letter grieved you, though only for a while. As it is, I rejoice, not because you were grieved, but because you were grieved into repenting. For you felt a godly grief, so that you suffered no loss through us.
Presumably, this is a reference to 1st Corinthians, which is a stark appraisal of the Corinthians. As usual with Paul, we have to read between the lines a bit to understand what his detractors were accusing him of. It appears he was accused of being only interested in winning a theological argument by hook or by crook, and not caring for the people of the church in Corinth.
Paul's defense against the charge is the passage quoted in the question and following in 2nd Corinthians 10:3-12 (ESV):
For though we walk in the flesh, we are not waging war according to the flesh. For the weapons of our warfare are not of the flesh but have divine power to destroy strongholds. We destroy arguments and every lofty opinion raised against the knowledge of God, and take every thought captive to obey Christ, being ready to punish every disobedience, when your obedience is complete.
Look at what is before your eyes. If anyone is confident that he is Christ’s, let him remind himself that just as he is Christ’s, so also are we. For even if I boast a little too much of our authority, which the Lord gave for building you up and not for destroying you, I will not be ashamed. I do not want to appear to be frightening you with my letters. For they say, "His letters are weighty and strong, but his bodily presence is weak, and his speech of no account." Let such a person understand that what we say by letter when absent, we do when present. Not that we dare to classify or compare ourselves with some of those who are commending themselves. But when they measure themselves by one another and compare themselves with one another, they are without understanding.
If you follow the logic, the phrase "we are not waging war according to the flesh" is there to emphasize that Paul has a no-holds-barred attitude to attacking false doctrine that threatens the Corinthians salvation. It's spiritual warfare in a sense, but the manifestation of the warfare is rational argument.
So it seems that reading Christian Pacifism into the text is just that. However, there is an argument to be made based on the first verse of the chapter which characterizes Jesus as meek and gentle. As Paul had previously told the Corinthians, "Be imitators of me, as I am of Christ." (1st Corinthians 11:1) So by extension, Paul may be suggesting that we, as Christians, remain meek and gentile in most circumstances.
Paul certainly knew of the story of Jesus cleansing the Temple, which appears neither meek nor gentle. But it's possible that Paul intends us to recall that moment in Jesus' life since He was uncharacteristically violent for the purposes of opposing unprofitable doctrine, injustice, and corruption. If Paul is suggesting what we call Christian Pacifism, it is a somewhat nuanced position.
While this particular passage can be read to support Christian Pacifism, it's certainly not the main point of Paul's argument here.