Actually, there is no conflict between Natural Law and the Bill or Rights. A Right to self defense against predators? A Right to keep anything you make from your own labor? A Right to say and believe as you want?
If anything, God's laws are in conflict with the Bill of Rights. How can you have a "freedom of religion" if your Judeo-Christian God has stipulated that "thou shalt have no other Gods before me".
You are trying to claim a dichotomy where none exists.
That’s why I choose Him over political identity or constitution