Syncretic religions (based on combining separate traditions) are a big part of the backdrop in Dune, with the Zensunni as the most obvious example. FH was probably inspired by the religious turmoil of the period, when new religions, sects, cults and spiritual movements were springing up almost daily. Scientology is perhaps the most famous one to have survived (and FH was obviously familiar with it, given its science fiction origins and link with Analog magazine), but others such as Rastafari, Wicca and Nation of Islam were also relatively recent religions or sects that were increasing their profile at the time. (The Rastafari movement in particular has intriguing parallels to Dune, with a living Emperor - who started out as a Duke - being deified by a poor and oppressed people, but the timeline makes it unlikely that this religion specifically was a significant influence.)
Many of these modern religious movements were syncretic, such as Theosophism, Wicca, Rev. Moon's Unification Church, Unitarian Universalism, Baha'i (though its adherents deny it) and the spiritual movement later known as New Age. The thought of unifying the major world religions (or forming new, strange mixtures of them) was natural in that context.
The main inspiration for the CET itself is no doubt the 1962-1965 (i.e. contemporary with the writing of Dune) ecumenical council of the Roman Catholic church known as Vatican II. It was, in the words of Tom Lehrer, a "big news story of the year," certainly to someone of Herbert's Catholic upbringing. In the statements attributed to CET delegates, we can hear echoes of Pope John XXIII's opening address to the Second Vatican Council:
What is needed at the present time is a new enthusiasm, a new joy and serenity of mind in the unreserved acceptance by all of the entire Christian faith, without forfeiting that accuracy and precision in its presentation which characterized the proceedings of the Council of Trent and the First Vatican Council. What is needed, and what everyone imbued with a truly Christian, Catholic and apostolic spirit craves today, is that this doctrine shall be more widely known, more deeply understood, and more penetrating in its effects on men’s moral lives. What is needed is that this certain and immutable doctrine, to which the faithful owe obedience, be studied afresh and reformulated in contemporary terms. For this deposit of faith, or truths which are contained in our time-honored teaching is one thing; the manner in which these truths are set forth (with their meaning preserved intact) is something else.
The liberal ideas and practices produced by the reformist and modernizing Vatican II council proved controversial within Catholicism, just as the OC Bible proved in Dune.
More generally, Vatican II was just the latest in a long line of church councils, some of which did indeed lead to riots and charges of heresy, as representatives from disparate strands of Christianity tried to come to agreement on theology and ritual. The most famous is the First Council of Nicaea, which produced the Nicene Creed defining what it means to be Christian. FH was probably familiar, at least to this extent, with this part of church history.
Finally, I would speculate that the Commission of Ecumenical Translators was inspired in part by the various Biblical translators, particularly the committees who did the King James translation, a work that was unpopular in its own time but came to be seen as definite. The CET Commentaries may also be an indication of a Talmudic template, which brings to mind the legendary 70 scholars (the inspiration for the CET's "Fourteen Sages"?) who produced the Septuagint, translating the Old Testament into Greek.
So, are these the most likely inspirations, or are there others I have missed?