Two bases:
First, the Constitution specifically prohibits the use of a religious test for any office.
Second, the Constitution also sets forth the qualifications for President (natural born citizen, age 35+, 14+ years residence in the United States), and does not grant Congress the power to add additional qualifications/restrictions.
"The Constitution is not a suicide pact" is a phrase in American political and legal discourse. The phrase expresses the belief that constitutional restrictions on governmental power must be balanced against the need for survival of the state and its people. It is most often attributed to Abraham Lincoln, as a response to charges that he was violating the United States Constitution by suspending habeas corpus during the American Civil War. Although the phrase echoes statements made by Lincoln, and although versions of the sentiment have been advanced at various times in American history, the precise phrase "suicide pact" was first used by Justice Robert H. Jackson in his dissenting opinion in Terminiello v. Chicago, a 1949 free speech case decided by the U.S. Supreme Court. The phrase also appears in the same context in Kennedy v. Mendoza-Martinez, a 1963 U.S. Supreme Court decision written by Justice Arthur Goldberg.
Thomas Jefferson offered one of the earliest formulations of the sentiment, although not of the phrase. In 1803, Jefferson's ambassadors to France arranged the purchase of the Louisiana territory in conflict with Jefferson's personal belief that the Constitution did not bestow upon the federal government the right to acquire or possess foreign territory. Due to political considerations, however, Jefferson disregarded his constitutional doubts, signed the proposed treaty, and sent it to the Senate for ratification. In justifying his actions, he later wrote: "[a] strict observance of the written law is doubtless one of the high duties of a good citizen, but it is not the highest. The laws of necessity, of self-preservation, of saving our country when in danger, are of higher obligation. To lose our country by a scrupulous adherence to the written law, would be to lose the law itself, with life, liberty, property and all those who are enjoying them with us; thus absurdly sacrificing the ends to the means."
But was that intended to apply to Muslims? What was the intended function of that clause? Why is it there?
and does not grant Congress the power to add additional qualifications/restrictions.
It does not specify that the President cannot be a retard. It does not specify that the President must be human. These are regarded as common sense or "axiomatic" stipulations of the office, needing no explicit articulation.
I argue that "Islam" is also one of these "axiomatic" exclusions. That it does not have to be specified because it isn't comprehended by the intent of that "religious test" clause.