What is the short list of principles that you say comprises such a core, and what part of the old testament do you say that list directly derives from?
Your question is insightful, and it strikes at the heart of the problem involved in the compromises that created the American Republic. While the sort of list you want could be compiled, it would have to be defended historically, not theologically or logically. Thomas Jefferson and Benjamin Franklin would have quite different views of that “common core” than Witherspoon, the Presbyterian minister and president of Princeton, for example.
It is simply impossible to deny that the Founding Fathers worked with moral presuppositions and belief in a Creator. In that respect, they differ radically from modern secular liberalism. We can say, correctly so, that the ACLU wants to create a nation which even the most “liberal” of the Founding Fathers would not recognize.
On the other hand, the Founders included Roman Catholics, Deists, people who not long afterward were described as Unitarians, some people whose personal religious convictions were not terribly strong, and a wide variety of mutually incompatible forms of evangelical Christianity, as well as groups such as the Quakers which would today be far outside the mainstream of evangelical Christianity but probably can be correctly described as evangelicals in the context of the 1700s. While none of the Founders were themselves Jewish, it's quite clear that Jewish leaders in America included some strong supporters of the Revolutionary War and that during or not long after the Revolution, nearly all of the states removed religious tests such as affirmation of the Trinity which had historically been used to bar Jews from civil office in England while allowing other dissenters such as Presbyterians and Congregationalists to retain their noble titles and in some cases hold other civil offices.
The Unitarians, Deists, and other “freethinkers” of the 1700s and 1800s believed there was a common core of moral values which could be derived from the Old Testament even if they didn't view it as inspired and inerrant Scripture.
We do not have that consensus today.
Historically, it's possible to point to a common core of civil values and morals that evangelical Protestants, conservative Roman Catholics, Orthodox Jews, and the older type of liberalism shared. Examples include the Second Table of the Law, with its warnings against such vices as lying, stealing, adultery, murder, coveting, etc., and the many related sins that connect to those commandments and which have civil laws and punishments which the Westminster Confession of Faith says are given in the Old Testament as examples for modern governments in their “general equity.”
For conservatives who believe in original intent, we need to recognize that some important changes happened between Plymouth Rock and the Constitutional Convention. While the first civil law code of Massachusetts included very detailed Scripture references for its laws and punishments, and it can fairly and correctly be said that the early colonies of New England were explicitly Christian, that cannot be said of the United States Constitution.
I'd like to say that America is a Christian Republic. I simply cannot say that based on the evidence of the original intent of the Founders.
Let's take the first and second commandments, for example. Can any of us seriously argue that Franklin and Jefferson would not have been severely persecuted and expelled, if not executed, for their views under Puritan practices of the 1600s, whether in New England, Scotland, or England itself? The charges would at the very least have included blasphemy and irreligious writings.
Whatever I might like to say about America's Christian heritage has to be tempered by the reality of what the Founders intended, and I simply do not believe we can make the claim that America is a Christian Republic without qualifying that so much that the term “Christian” is no longer defined in a biblical manner. To say America is founded in Judeo-Christian principles is a claim which can be defended historically.
I hope that's of some help.