Tom Van Dyke's citing Daniel Dreisbach's understanding of religion & the original Constitution that responds to Kramnick and Moore's "The Godless Constitution" thesis is fair, as far as it goes.
However I find Dreisbach's mantra -- religion is left to the states -- unsatisfactory in the sense that it avoids the natural rights framework of the American Founding. The natural rights of conscience were the most "unalienable" of rights intimated in the text of the Declaration of Independence. Under the "religion was left to the states" rule (indeed, as originally conceived, the overwhelming MAJORITY of things were left to the states that our modern, post 13th, 14th Amendment AND post Wickard v. Filburn understanding of the Commerce Clause world finds anathema) state and local governments get to ride roughshod over many unalienable rights of conscience.
This is, some might not like to hear it, EXACTLY like "slavery was left to the states." And that's because, as with religion, it was. BOTH issues involve fundamental matters of unalienable rights that states originally were permitted to violate, as per federalist compromises.
Now, the hard question that follows, with regard to religion, is, what practices did the original US Constitution permit that violated the natural, unalienable rights of conscience? Prohibiting anyone from openly professing their religious beliefs and practicing them in a peaceful manner that didn't contravene the secular civil law? Yes, some states did that, and virtually everyone agrees THOSE things, originally constitutional according to the "religion is left to the states" rule, violated the natural rights of the DOI. They are today, understandably, held to be unconstitutional.
But there are harder questions. What if I am free to practice my religion under the confines of the secular civil law, but am barred from holding public office because I am not of the "right" religion of the state? What if the state uses my tax dollars to support a sectarian religion in whose doctrines I don't believe?
Answering these questions is the matter of not just one book, but many. So I'll pick my battles, one at a time and carefully. Regarding sectarian religious tests, the original Constitution permitted states to enact them. But, at the very least, Ben Franklin, Benjamin Rush, Richard Price, and many others thought they violated the natural rights of conscience that the Declaration of Independence protected.
For instance, what follows is PA's original Constitution which Ben Franklin helped pen. It contains a religious test that violated the natural rights of the Declaration of Independence. It held:
And each member, before he takes his seat, shall make and subscribe the following declaration, viz:
I do believe in one God, the creator and governor of the universe, the rewarder of the good and the punisher of the wicked. And I do acknowledge the Scriptures of the Old and New Testament to be given by Divine inspiration.
Franklin despised the "un-Declarational" religious test contained therein. (He had to accede to such a test as a political compromise.) As he wrote in a letter to John Calder:
I agreed with you in Sentiments concerning the Old Testament, and thought the Clause in our Constitution, which required the Members of Assembly to declare their belief that the whole of it was given by divine Inspiration, had better have been omitted. That I had opposed the Clause but being overpower'd by Numbers, and fearing might in future times be grafted on [it, I Pre]vailed to have the additional Clause that no [further or more ex]tended Profession of Faith should ever [be exacted. I ob]serv'd to you, too, that the Evil of it was [the less, as no In]habitant, nor any Officer of Government except the Members of Assembly, were oblig'd to make that Declaration. So much for that Letter. To which I may now add, that the[re are] several Things in the old Testament impossible to be given by divine Inspiration, such as the Approbation ascrib'd to the Angel of the Lord, of that abominably wicked and detestable Action of Jael the Wife of Heber the Kenite. If the rest of the Book were like that, I should rather suppose it given by Inspiration from another Quarter, and renounce the whole.
Benjamin Rush -- originally an orthodox Christian, but later converted to universalism, believing all will eventually be saved -- likewise despised PA's religious test. Here are excerpts from two of his letters to English Whig Richard Price (who in turn greatly influenced our key Founders).
In the first, Rush calls such a test "a stain from the American Revolution."
[15 Oct. 1785]
I took the liberty of publishing, with your name, your excellent letter on the test law of Pennsylvania. It has already had a great effect on the minds of many people, and I doubt not will contribute more than anything to repeal that law. Dr. Franklin, who has succeeded Mr. Dickinson as our governor, has expressed his surprise at the continuance of such a law since the peace, and we hope will add the weight of his name to yours to remove such a stain from the American Revolution.
And here he notes that such test was eventually repealed:
[22 Apr. 1786]
I am very happy in being able to inform you that the test law was so far repealed a few weeks ago in Pennsylvania as to confer equal privileges upon every citizen of the state. The success of the friends of humanity in this business should encourage them to persevere in their attempts to enlighten and reform the world. Your letter to me upon the subject of that unjust law was the instrument that cut its last sinew.
A "stain from the American Revolution." Perhaps not "unconstitutional" according to 1791's amended Constitution. But clearly, "un-Declarational."