Founding Fathers and Islam:

This page from the Library of Congress shows that many of the Founding Fathers were quite tolerant of Islam. The page also demonstrates the concept of the US being "founded" on "Judeo-Christianity" (that is, some type of special status given to Christians and Jews, but excluding other religions) is without historical merit.

Many believed Protestant Trinitarian Christianity only ought to have "rights." And the folks who tolerated/granted rights to Jews, tended to tolerate/grant rights to Muslims, Hindus, Catholics, Deists and Unitarians along with them (with a whole variety of opinions in between -- for instance the link notes that Locke believed in tolerating Islam, but wouldn't extend toleration to atheists or Catholics).

To theistic rationalists like Jefferson, Adams, Franklin, and I'd argue Washington and Madison as well, Muslims worshipped the same God as Christians and Jews. And since their religion inculcated morality, it was "sound" like Judaism and Christianity. To the theistic rationalists, whether religion produced morality was the test of its validity, not adherence to any kind of "creeds" (like the Nicene or Anathasian) or doctrines like the Trinity, Incarnation, and Atonement. This is why the religion of the key Founders arguably is not "Christianity" even though clearly it was strongly influenced by Christianity.

Hutson's link catches on to this dynamic:

In 1783, the president of Yale College, Ezra Stiles, cited a study showing that "Mohammadan" morals were "far superior to the Christian." Another New Englander believed that the "moral principles that were inculcated by their teachers had a happy tendency to render them good members of society." The reference here, as other commentators made clear, was to Islam's belief, which it shared with Christianity, in a "future state of rewards and punishments," a system of celestial carrots and sticks which the Founding generation considered necessary to guarantee good social conduct.

Indeed, to our key Founders, that's what "sound religion" was all about.