"Servants" in the KJV usually means "slaves." The Americans actually pushing for independence were not slaves, so that verse is irrelevant to the topic. "Render unto Caesar the things that are Caesar's" isn't the whole verse; it ends with "and unto God the things that are God's." Sometimes people focus so much on the first part of that verse that they forget the importance of the second part, which clearly points out that Caesar's domain isn't the only domain there is. Another question from that verse: Did "Caesar" (or, in this case, the British Empire) have a right to the things it was taking? If not, the verse doesn't apply.
And the Americans did not just decide that they were tired of British rule for no reason. They were reacting to policies constricting the rights of self-rule that they were accustomed to. It was a very conservative revolution in that sense, which is why Edmund Burke supported it while later opposing the French Revolution. The Americans were being reactionaries more than true revolutionaries; the British were the real revolutionaries in this case.
The Americans also remembered the House of Stuart (Patrick Henry referenced it in his famous "Liberty or Death" speech, when a few restrictions on rights did eventually lead to a lot of death. How far would the British go? They couldn't be sure. If they allowed the British to go far enough that people's lives really were in danger, by then resistance would probably be too late. How far is K.C. willing to carry the non-resistance thing? Does K.C. think the Jews had a good idea in not fighting back against Hitler until he was literally trying to kill them all? Did it work? What about all the wars in the Bible? Should the Israelites have practiced non-resistance to the Philistines unless the Philistines were trying to kill them? Would that have worked?
The War for Independence is admittedly a sticky issue, but the Americans in general tried to do things peaceably until the first shot was fired (and no one knows by whom). The war started by accident. At that point, if they had not fought, then the Americans leaders, at least, would have been tried in England and probably hanged. So it may not have started as a life-and-death issue, but after the first (accidental) shot, it became one.
That's kind of a muddle of thoughts, but hopefully something in there will help.