Just a brief comment.
When Jesus said, "Give to Caesar what is Caesar's, and to God what is God's" he was taking a major swipe at the legitimacy of Caesar as a living god. Everything belongs to God as the Creator owns the Creation. Jesus was trying to make the point that Caesar is demanding worship, a false idol.
If that were the case, he would have told them to give Ceasar nothing at all. That would break the 1st commandment.
Jesus is separating the Church and State. (not abolishing it completely, but making the 2 distinct and separate) In many ways the Jewish state was a corrupt theocracy at the time, and was operated under an empire that wanted to press its will on the Jews. What you are suggesting is exactly the trap that they wanted to put Jesus into. In other words. Jesus takes a LIBERTARIAN stance here.

Now this bit is QUITE inconvenient to neo-cons.
Matthew 22
Then the Pharisees went out and laid plans to trap him in his words. 16 They sent their disciples to him along with the Herodians. “Teacher,” they said, “we know that you are a man of integrity and that you teach the way of God in accordance with the truth. You aren’t swayed by others, because you pay no attention to who they are. 17 Tell us then, what is your opinion? Is it right to pay the imperial tax[a] to Caesar or not?”
18 But Jesus, knowing their evil intent, said, “You hypocrites, why are you trying to trap me? 19 Show me the coin used for paying the tax.” They brought him a denarius, 20 and he asked them, “Whose image is this? And whose inscription?”
21 “Caesar’s,” they replied.
Then he said to them, “So give back to Caesar what is Caesar’s, and to God what is God’s.”
22 When they heard this, they were amazed. So they left him and went away.
phar·i·see/ˈfarəsē/
Noun:
A member of an ancient Jewish sect, distinguished by strict observance of the traditional and written law, and commonly held to have pretensions to superior sanctity. (In other words, these were the high-horsed evangelicals of the day that influenced government aka neocons)