The First Amendment was written by the Founders to protect the free exercise of Christianity. They were making no effort to give special protections to Islam. Quite the contrary….
Islam has no fundamental First Amendment claims, for the simple reason that it was not written to protect the religion of Islam. Islam is entitled only to the religious liberty we extend to it out of courtesy. While there certainly ought to be a presumption of religious liberty for non-Christian religious traditions in America, the Founders were not writing a suicide pact when they wrote the First Amendment.
Actually, both the First Amendment and the No Religious Test Clause of the original Constitution were quite deliberately written to cover all religions. Many state constitutions of the era did limit their protection to Protestants (New Jersey, North Carolina, and Vermont) or Christians (Delaware, Maryland, and Massachusetts). Some others (New Hampshire and South Carolina) provided for funding of Protestant or Christian teaching, or more broadly established Protestantism, but did not limit religious freedom protections or office-holding.
But the U.S. Constitution did not have any such limitation. James Iredell, later one of the first Justices of the Supreme Court, specifically defended the No Religious Test Clause on precisely these grounds:
I consider the clause under consideration as one of the strongest proofs that could be adduced, that it was the intention of those who formed this system to establish a general religious liberty in America….
But it is objected that the people of America may, perhaps, choose representatives who have no religion at all, and that pagans and Mahometans may be admitted into offices. But how is it possible to exclude any set of men, without taking away that principle of religious freedom which we ourselves so warmly contend for? This is the foundation on which persecution has been raised in every part of the world. The people in power were always right, and every body else wrong. If you admit the least difference, the door to persecution is opened.
Likewise, Thomas Jefferson wrote that the Virginia statute protecting religious freedom, which he drafted, deliberately covered “the Jew and the Gentile, the Christian and Mahometan, the Hindoo, and Infidel of every denomination,” and that a proposal to mention Christ in the bill “was rejected by a great majority, in proof that they meant to comprehend” all religions. Barnes v. Inhabitants of First Parish in Falmouth, 6 Mass. 401 (1810), expressly stated that the Massachusetts Constitution’s general protection for religious freedom — similar in this respect to the First Amendment’s general protection — “secured liberty of conscience, on the subject of religious opinion and worship, for every man, whether Protestant or Catholic, Jew, Mahometan, or Pagan.” (Massachusetts limited various offices to Christians, but otherwise provided for religious freedom without such a limitation.)
I know of no sources that suggested that anyone during the Framing era understood the Constitution as excluding “Mahometans,” or non-Christians more generally, from either the Free Exercise Clause or the No Religious Test Clause. (The Framers were open to general religious references, and sometimes references to Christianity, in the speech of the federal government; they likely had a much narrower view of the Establishment Clause than that reflected in the Supreme Court’s modern caselaw. But that is a separate matter from which religions were protected by the Free Exercise Clause and the No Religious Test Clause.)