A number of years ago, a co-author and I tackled this issue which you can view here.
America's Founders often used the term "religion" and when they did it's a mistake to conclude they meant "Christianity" to the exclusion of other religions like Islam. So if "religion" is granted rights and has restrictions placed on it, such applies to "religion" in general. Islam as a religion therefore gets equally protected under such principles as any form of Christianity.
Below is a quote from one of the authors at the Immanent Frame:
Today we often refer to “Judeo-Christian civilization” but, as Spellberg points out, this term excludes Muslims from that shared history. Spellberg’s book reminds us of the strong tradition of tolerance in the United States, but also of how it is easy to fall short of that goal. . . .This is true. However, the actual history including both laws and social institutions is a bit more complicated. Yes, there was a remarkable degree of theoretical liberality and ecumenicism that saw Islam being given equal rights under the auspices of protecting "religion." Judaism and all of the various forms of Christianity, orthodox, unorthodox, whatever we might debate qualifies as "real Christianity" were with Islam, "religions."
There were also at the state level different ways of dealing with religion that varied by state. Roman Catholics, for instance, might have their full religious rights in one state, but not another.
If there was some kind of institutional zeitgeist, it was a preference for social Protestant Christianity as the "in" group. All others -- Roman Catholics, Jews, Muslims -- in the "out" group.
For instance, militant unitarians John Adams and Thomas Jefferson as formally and nominally connected to respectively the Congregational (Adams) and Anglican-Episcopal (Jefferson) Churches received cover under the auspices of Protestant Christianity's privileged social standing, along with those who actually devoutly believed in the orthodox creeds and doctrines to which those churches were grounded.