Well, I've facilitated discussions on Romans 13 and the American Founding that have almost beaten the horse to death. I've little dealt with Mark 12:17, where Jesus said, "Render to Caesar the things that are Caesar's, and to God the things that are God's. ..."
Let's do so now. The passage, like many of Jesus' sayings, has specific and broader meanings. In the specific sense, Jesus recognized Caesar's temporal governmental authority over Him and His followers. In a broader sense, believers are to be subject to the civil government, pay their taxes and so on. And because the civil government Jesus recognized was that of pagan imperial Rome, logic instructs Jesus advanced institutional separation of church and state, where Christians can live under a "state" that is a pagan and un-godly entity as was Caesar's.
This parallels Romans 13 as I understand it. Most biblical scholars agree that the higher power St. Paul instructed believers to submit to -- in the literal instance, the specific example ordained by God -- was the pagan psychopath Nero.
In a symbolic sense, that would set a very low standard for "rulers" to meet in order to properly maintain their Romans 13 power.
Now, I've heard some orthodox believers argue Paul really wasn't telling believers to submit to Nero but rather to some ideal "godly" government because he said "rulers are not a terror to good works, but to the evil," and that they are "minister[s] of God to thee for good." As such, Nero or any tyrannical ruler wouldn't qualify.
But, when one puts Romans 13 together with Mark 12:17 that interpretation seems less biblically sound. In the latter, Jesus recognizes the civil legitimacy of an imperial pagan entity, arguably a tyranny. (By this time Rome ceased being a "republic" and was now an "empire.")
When the American Founders posited the notion "rebellion to tyrants is obedience to God," because they operated in a general Christian context, they had to deal with the Bible.
That notion is not "biblical." Whether it's "anti-biblical" (something the Bible forbids) or "a-biblical" (something the Bible doesn't teach, but doesn't necessarily forbid) is debatable. What's not debatable, in my very learned opinion, is the notion "rebellion to tyrants is obedience to God," however nice it sounds, is NOT what the Bible teaches.
With Romans 13, while most of the "key Founders" ignored it (it ended up, with all of Paul's words, on Jefferson's cutting room floor), to satisfy the public (who may have been more concerned with the Bible) ministers like Jonathan Mayhew, Samuel West offered "reasoned" explanations for why Romans 13, properly understood, didn't forbid revolt against tyranny.
But the Founders still had Jesus to deal with on government. And, unfortunately for them, Jesus' words offered no satisfaction for their plans on how to deal with tyrannical rulers. Indeed Gouvernor Morris lamented to George Washington the insufficiency of Jesus' words in this respect.
However, Jesus did inspire them. Whether they were orthodox Christians, Arians or Socinians, they thought Jesus the "best person," regardless of whether He (or he) was 2nd person in the Trinity or simply a man on a mission from God to save us through his (not His) perfect moral example. Jesus always did the right thing. So on how to conduct your personal affairs, follow Jesus.
George Washington's 1783 Circular to the States (not written by GW, but given under his imprimatur) "dispose[d] us ... to do Justice, to love mercy, and to demean ourselves with that Charity, humility and pacific temper of mind, which were the Characteristics of the Divine Author of our blessed Religion, and without an humble imitation of whose example in these things, we can never hope to be a happy Nation."
But there was a figure (how much of his example was actual history, how much literary embellishment, I do not know) who, unlike Jesus, did rebel against tyranny. In fact, a contemporary of Jesus, he rebelled against the same tyranny to which Jesus submitted: Cato the Younger. He purportedly committed suicide (not a very Christian act) rather than submit to the tyranny of Caesar. He was, as it were, the last of the Roman republicans, after Rome transformed into an empire.
It was Joseph Addison's tragedy Cato that inspired the American Founders. Washington had this play performed for his "Christian" troops to inspire them.
Of course, if Jesus's words offered support for what Washington and the others did against the British they would have been used. But, alas, they didn't. So the Roman Stoic Cato filled in.