Would Bryan Fischer Consider Joseph Story a "Christian"?

No. And arguably the theological system to which Joseph Story refers reproduced below is not necessarily "Christianity" as Fischer understands the concept. See Ed Brayton's link.

Fischer wrote:

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.

Fischer misunderstands Story's quotation:

"Probably at the time of the adoption of the constitution, and of the amendment to it, now under consideration, the general, if not the universal, sentiment in America was, that Christianity ought to receive encouragement from the state, so far as was not incompatible with the private rights of conscience, and the freedom of religious worship. An attempt to level all religions, and to make it a matter of state policy to hold all in utter indifference, would have created universal disapprobation, if not universal indignation...

"The real object of the amendment was, not to countenance, much less to advance Mahometanism, or Judaism, or infidelity, by prostrating Christianity; but to exclude all rivalry among Christian sects, and to prevent any national ecclesiastical establishment, which should give to an hierarchy the exclusive patronage of the national government."

And Story's quotation, while interesting, is not determinative; the text of the Constitution is. And the text protects "religion" NOT "Christianity" only.

But we have to wonder what theological system Joseph Story even speaks of when he invokes "Christianity." Story himself, like many other elites during his time, seemed to believe in biblical unitarian-universalism and considered that true "Christianity." And, accordingly to such theology, Islam is a valid path to God.

Here is Story on what he DIDN'T believe about Christianity:

TO WILLIAM WILLIAMS, ESQ.

Washington, March 6th, 1824.

...The Unitarians are universally steadfast, sincere, and earnest Christians.

They all believe in the divine mission of Christ, the credibility and authenticity of the Bible, the miracles wrought by our Saviour and his apostles, and the efficacy of his precepts to lead men to salvation....They differ among themselves as to the nature of our Saviour, but they all agree that he was the special messenger of God, and that what he taught is of Divine authority. In truth, they principally differ from other Christians in disbelieving the Trinity, for they think Christ was not God, but in the Scripture language “the Son of God.”

And here is testimony from Story's brother, speaking to and through Story's son:

After my continued absence from home for four or five years, we met again, your father being now about eighteen years old, and renewed our former affection towards each other. At this time we were, from a similarity of sentiment, drawn more closely together. I allude particularly to our religious opinions. We frequently discussed the subject of the divinity and the humanity of Christ, and we both agreed in believing in his humanity. Thus you see that your father and myself were early Unitarians, long before the doctrine was preached among us by any one, unless I except Dr. Bentley of Salem.

In other words, Story was a Socinian Unitarian, believing Jesus was 100% human and not divine at all. And here is what Story thought on salvation:

This faith he retained during his whole life, and was ever ardent in his advocacy of the views of Liberal Christians. He was several times President of the American Unitarian Association, and was in the habit of attending its meetings and joining in its discussions. No man, however, was ever more free from a spirit of bigotry and proselytism. He gladly allowed every one freedom of belief, and claimed only that it should be a genuine conviction and not a mere theologic opinion, considering the true faith of every man to be the necessary exponent of his nature, and honoring a religious life more than a formal creed. He admitted within the pale of salvation Mahommedan and Christian, Catholic and Infidel. He believed that whatever is sincere and honest is recognized of God; — that as the views of any sect are but human opinion, susceptible of error on every side, it behooves all men to be on their guard against arrogance of belief; — and that in the sight of God it is not the truth or falsity of our views, but the spirit in which we believe, which alone is of vital consequence. [Bold mine.]