The Bible came from the Catholic Church, so the answer is easy there. If the Catholic Church has no authority, then the Bible has no authority.
That answers your question: "who are the arbiters of the bible?"
But, you had a preamble to the question: "If the US is governed by biblical principles..." I guess I will have to look at the chat below. I am one Catholic who likes the Separation of Church and State. I think it is good for both Church and State. I don't think a Christian Theocracy was established.
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