Further, I think the first tablet of the 10 Commandments is "special" revelation and doesn't count. [Freedom of religious conscience as "Freedom to sin" does not obtain.]
Natural law arguments, "general" revelation, don't need the Bible be derived. When the Bible agrees with reason in natural law arguments [James Wilson and many or most in the Founding era believed they were always in harmony], that doesn't make the arguments "theocratic," i.e., beyond reason, and rejected out of hand.
This covers a key point I often stress. "The laws of Nature and Nature's God" -- the metaphysical grounding for America's Founding ideals -- is not shorthand for what's written in the Bible, but rather what's discovered by reason. The Christian natural lawyers, as TVD pointed out, believed the two wouldn't contradict one another because they ultimately derived from the same source.
But because the Founders were trying to take sectarian disputes out of politics, they had to leave those parts of special revelation that couldn't be confirmed by general revelation (natural law) out of politics. Hence America's was founded on the proposition that the second tablet is a proper source of public law (because it's part of the natural law) but not the first.
Jews, Christians, Muslims, Unitarians, Hindus could all agree on the norms in the second tablet (don't murder, don't steal, don't bear false witness) but the first tablet as part of "the Law" is what puts them at their throats in the political-theological wars. It was the first tablet as Law that got Servetus burned at the stake in Calvin's Geneva. Currently it's the first tablet as Law that limits the Christians' freedom to convert Muslims under Sharia.
So whereas the American Founding doesn't stand for the proposition that men necessarily have a right to sin according to general revelation (the natural law that all good men of all religious faiths could determine from reason) it does stand for the proposition that men have a right to sin according to special revelation. It has to; you cannot get religious liberty for all or even between Trinitarians and non non-Trinitarians without it.
Not every, perhaps not even most unitarians of the Founding era were free wheeling Jeffersonian Epicurians. Some were quite pious and they thought Trinitarianism a grave sin -- a violation of the First Commandment; worshipping Jesus as a false god takes glory away from the Father.
And again whether our natural rights are limited to what the natural law permits is another, much harder proposition to tackle. I noted Randy Barnett's article arguing natural rights are not limited by what natural law ethics permit. For the opposite point of view see Philip Hamburger's article on the matter.
Finally, to drive the point home, here is a classic post by Eugene Volokh that shows how obvious it is that granting religious liberty to Hindus -- something all "key Founders" in principle believed in, even if they didn't get a chance to see the results -- necessarily means giving them a right to break parts of special revelation (i.e., the Bible):
Say that a few Hindus are hired as teachers in a public school district; and that some people start to complain. Hindus, they point out, routinely and unabashedly violate three of the Ten Commandments (they worship other Gods, they create images of their Gods, and they don't observe the Sabbath). What's more, the Hindus would therefore be bad role models for children: Some kids, seeing the teachers' example, might be drawn towards Hinduism; and other kids, seeing some nearby authority figures who aren't Christian, might have their belief in Christianity undermined -- and of course the results of that would be truly dire, since they would jeopardize the children's salvation. Therefore, the people argue, the school must refuse to hire Hindu schoolteachers.
My guess is that such an argument would be pretty broadly condemned, even by many conservatives and Christians (and for that matter conservative Jews and members of other religions; I focus on Christians here simply because their views are especially salient in American public debates). Religious freedom, those people would point out, means (among other things) that we tolerate religious differences, and that we don't discriminate against people in government employment just because of their religious beliefs.
We may earnestly believe that they're wrong -- whether they're non-Christians, heretics, apostates, agnostics, atheists, or what have you. We may believe that they'll go to Hell for their errors (though we may sincerely regret that). We may want our children not to make these errors. But we ought not legally punish people, or deny them access to jobs and other government benefits, because of their violations of certain religious laws, even some of the laws in the Ten Commandments. (I'm sure that some people don't take such a tolerant view, but I believe that many conservative Christians would quite sincerely endorse it -- I certainly know some such people personally.)
Of course, this hasn't always been so: Historically, religious discrimination, intolerance, and persecution has been the rule rather than the exception; and even in the U.S., various groups -- Catholics, Jews, atheists, and others -- have in the past faced substantial governmental discrimination, though generally less than in other countries of the time. But today, the general view, again, seemingly shared by a broad range of people, including many devout, conservative Christians, is that toleration is the more just approach. And, in particular, this means that
1. People's failure to obey religious laws -- even three of the Ten Commandments -- is not by itself reason enough to punish them, or deny them equal access to government benefits.
2. The risk that others will follow this bad example is also not reason enough to punish the violators of religious laws (here, the Hindus), even if we sincerely believe that following the example will lead to eternal damnation.
3. Some religious laws, including some of the Ten Commandments, are matters to be enforced not by man but by God.
I should note, according to the First Amendment and (private anti-discrimination law theory) Hindu Americans are entitled to more than just toleration but a "right" to be a practicing Hindu.
As Volokh noted, Hindus and other sects didn't always, in practice, have equal rights in America (mainly at the state and local level). But the rhetoric of every "key Founder" suggests Hindus had an unalienable natural right to be practicing Hindus, even though such clearly breaks God's law in special revelation.