We don't know who first made this argument, but it was popularized by anti-separation activist David Barton in his 1992 book, The Myth of Separation. On page 41 of Myth he argues as follows:
True to form, Barton's assertion contradicts virtually everything we know about the Danbury letter. First, there is no evidence that the Danbury Baptists wrote to Jefferson because of a rumor that "a particular denomination was soon to be declared a national denomination." Barton does not evidence this claim, and the argument is implausible on it's face: the First Amendment had been in effect for about a decade, and it was universally understood that Congress had no ability to declare a national religion (see Thomas Curry, The First Freedoms, ch. eight; Leonard Levy, The Establishment Clause, chs. 4-5). It is difficult to believe, in other words, that any group of well informed citizens--let alone Baptists, who were generally knowledgeable on issues of religious freedom--would have taken such a preposterous rumor seriously.
Second, a good deal of Jefferson's correspondence with religious groups during his presidency is extant, and nowhere in this correspondence do we find Jefferson addressing rumors of a national religious establishment. If a national establishment was the context of the Danbury letter, the Danbury Baptists were, so far as we can tell, alone in that concern.
Third, and more important, a copy of the Danbury Baptist's letter to Jefferson survives, and it utterly contradicts Barton's reading. The letter does not mention a national establishment; rather, the letter is concerned with the lack of religious liberty Baptists enjoyed in the state of Connecticut. The Baptist complaint was that the Connecticut state constitution did not prohibit the state from legislating about religious matters. As a consequence, they argued, "...what religious privileges we [Baptists] enjoy (as a minor part of the state) we enjoy as favors granted, and not as inalienable rights: and these favors we receive at the expense of such degrading acknowledgements as are inconsistent with the rights of freemen."
The "degrading acknowledgements" referenced here refers to a system of religious taxation that forced many Connecticut Baptists to support the established Congregationalist church. According to church/state scholar Derek Davis, Connecticut law allowed the Baptists to rout their religious taxes to their own churches, but this involved locating and filling out an exemption certificate, and many Connecticut communities either made it difficult to obtain the certificates, or refused to approve the exemptions once submitted (see, "What Jefferson's Metaphor Really Means," Liberty, Jan/Feb, 1997, p. 13). Beyond this, the Baptists found the law unjust and discriminatory in that it favored Congregationalism over other denominations. According to Davis, the Connecticut Baptists began a petition campaign in 1800 to put pressure on the state legislature to rescind the tax. The letter to Jefferson appears to have been a part of that campaign:
In the third paragraph of the letter the Baptists observe that Jefferson "is not the national legislator" and that "the national government cannot destroy the laws of each state," further indicating that the concern of the Danbury Baptists was the state's Congregationalist establishment. Their hope, apparently, was that Jefferson might use both his own moral influence as a beloved founder and the bully pulpit of his office to convince the Connecticut legislature to rescind the establishment:
Ironically, in his 1996 book, Original Intent, Barton rejects his 1992 interpretation of the Danbury letter. Nowhere in the book does he reference any rumor about a national establishment. Rather, Barton now argues the Baptists first wrote Jefferson to express their concern that the First Amendment might be interpreted to allow Congress to regulate religious expression! This view, over course, is absurd, and we refute it here.
In summary, there is no evidence that the context of the Danbury letter was a rumor of a national establishment. On the contrary, the concern of the Danbury Baptists was religious oppression in the state of Connecticut. Jefferson used the letter as an opportunity to express his own views that the First Amendment created a "wall of separation between church and state." This was no mere assurance that Congress could not establish a national religion. It was a response to the very thesis of the Baptists' letter: that religious rights are by nature inalienable. The Baptists wanted that view to prevail in Connecticut. Jefferson's metaphor assured them that this was already true on the national level, and that the federal government had no right to legislate on religious matters in any way.