There are at least two problems with the way the question is framed. One, it's a false dichotomy; there were more than two ideological sources. Harvard historian Bernard Bailyn also names "Whig," "Common Law" and "Greco-Roman."
Secondly, the sources bleed into one another (that is, they aren't mutually exclusive). John Locke was a "Whig," a man of the Enlightenment, and called himself a "Christian." George Washington's virtues were arguably consistent with both "Judeo-Christianity," and "Greco-Romanism." GW was as much a "Stoic" as a "Christian."
Further, there will never be any kind of "settled" answer among men as to what's an authentic "Christian" tenet that distinguishes itself from an "Enlightenment" tenet. I hate to sound like a deconstructionist, but essentially this is a continuing "discursive" process.
Sorry to further pick on the men, but David Barton and Peter Marshall illustrate the false dichotomy from the "Christian right" perspective. As Peter Marshall noted:
Research has revealed that Enlightenment philosophy was far less influential in the thinking of the Founding Fathers than has been taught in recent decades.
He noted this as he misunderstood the "research" that supposedly supported his point.
David Barton has likewise tried to paint "Enlightenment" as Hume, Voltaire, Rousseau and whatever presented itself as either non-Christian or least identifiably "Christian" so as to "capture" the Founding for "Christian" sources.
Well, the American Founding, the "republicanism" thereof, and the Enlightenment the Founding Fathers followed presented themselves as compatible with and often under the auspices of "Christianity." Likewise Jefferson, J. Adams, Franklin and others presented their creed as "Christianity" not Deism.
Presenting something under the auspices of "Christianity" is instructive of the "history of Christendom" -- the good with the bad, the orthodox with the heresies -- but tells us little about the "mere Christianity" that draws lines over which there is reason to argue in the first place.
What's there to fight about if "Christianity" includes Calvin, Pat Robertson, Mormonism, the French Revolution, churches that perform same sex marriages and even self proclaimed atheists and witches? All of these things have presented themselves under the auspices of "Christianity."
As it relates to the Founding, the "Enlightened" Christians of the American Founding and their philosophical heroes like the aforementioned J. Adams, Jefferson and Franklin, Jonathan Mayhew, Charles Chauncy, Joseph Priestley, Richard Price, John Locke, Isaac Newton disproportionately embraced the Arian and Socinian heresies. Socinianism predates the Enlightenment and Arianism traces back to fourth century.
How authentically "Christian" are the Arianism and Socinianism that the "Christians" of the Enlightenment tended to embrace because they viewed said as more "rational" than the Trinity?
Likewise, Locke's understanding of Romans 13 that held men had a "right" to rebel against tyrants (when the text of the Bible says no such thing) that the unitarian Jonathan Mayhew and many trinitarian preachers followed. Is it authentically "Christian"? If not, is it "Enlightenment"? Is it both?