In pre-Christian thought, Plato and his Theory of Forms is the most obvious antecedent of C.S. Lewis's theory of Transposition. This likely made its way into Christianity especially through the influence of the Neoplatonists. However, I will leave it to someone with more specific knowledge of Plato and Neoplatonism to trace that influence on Christianity in general, and on C.S. Lewis in particular.

A more recent precedent, within Christian thought, is Emanuel Swedenborg (1688–1772) and his Doctrine of Correspondence.

Platonism, Neoplatonism, and Swedenborg

Swedenborg himself undoubtedly drew on Plato's theory of forms in conceiving and formulating his doctrine of correspondence. Some contemporary scholars consider Swedenborg to be a Neoplatonist, based in part upon the clear kinship between Swedenborg's doctrine of correspondence and Plato's theory of forms. See, for example, "Jung and Swedenborg: modern Neoplatonists," by Mats Winther.

Swedenborg is also known to have read at least one Neoplatonic work: Pseudo-Aristotle, On the More Secret Part of Divine Wisdom according to the Egyptians. The last regular entry in Swedenborg's notebook of quotes from and commentary on various classical and contemporary thinkers, published as A Philosopher's Notebook (Philadelphia, PA: Swedenborg Scientific Association, 1931, p. 508-511) is a laudatory notice of this particular Neoplatonic work.

Swedenborg's Doctrine of Correspondence

However, as pointed out in the above paper by Mats Winther, and in my discussion about it with the author on Reddit here, Swedenborg developed the concept of a "correspondence" between spiritual and material things into a much more complex and sophisticated doctrine than the more general one that existed in Plato's philosophy and among the Neoplatonists.

In Swedenborg's doctrine of correspondence, every material object, and even every material force and action, down to its tiniest detail, is a precise expression on the physical level of some corresponding spiritual object, force, or action. While it may lack what we might call the "resolution" of its spiritual correspondent, at the lower level of "resolution" possible on the material level of reality, it is a completely faithful representation of the spiritual entity that is its correspondent and cause.

Here is one of Swedenborg's own definitions of "correspondence" as he uses the term:

First, I need to state what correspondence is. The whole natural world is responsive to the spiritual world—the natural world not just in general, but in detail. So whatever arises in the natural world out of the spiritual one is called "something that corresponds." It needs to be realized that the natural world arises from and is sustained in being by the spiritual world, exactly the way an effect relates to its efficient cause. (Heaven and Hell #89)

This concept—that everything in the natural, or material world reflects, not just generally, but in detail higher, spiritual realities—is so fundamental to and pervasive in Swedenborg's theology that he uses it not only to explain how God forms both the spiritual and material universes as finite expressions of the infinite nature of God, but also to interpret the Bible as a book in which God has expressed higher, spiritual realities in material world stories, events, and metaphors. Correspondence is also the key to Swedenborg's solution for the classic "mind-body problem," or how the human mind relates to and expresses itself in the body.

C.S. Lewis and Swedenborg

It is unknown (to me, anyway) just how much of Swedenborg's works, and thought, C.S. Lewis read, and was aware of. However, Lewis mentions Swedenborg several times in his writings, so we know at minimum that Lewis had some knowledge of Swedenborg. Further, Swedenborg is considered one of the primary sources of Lewis's book The Great Divorce (see the "Sources" section on its Wikipedia page).

Given how central the doctrine of correspondence is to Swedenborg's system, it is highly unlikely that Lewis did not have some familiarity with it. It very likely that even though Swedenborg did not use the word "transposition," Swedenborg's doctrine of correspondence was one of the sources for Lewis's idea of transposition.

In fact, in his sermon or essay "Transposition" (available online in a Gutenberg text version here), Lewis several times uses Swedenborg's term, "correspondence," in describing the relationship between higher and lower order realities that he calls "transposition."

It should be said, however, that despite fairly clear lines of influence of Swedenborg on C.S. Lewis, Lewis did not consider Swedenborg's claimed spiritual-world experiences to be genuine, and remained skeptical and dismissive of Swedenborg's overall theological system.

"Correspondence" in Lewis's essay on Transposition

In fact, reading Lewis's "Transposition" essay from the perspective of a Swedenborg scholar, I suspect that Lewis was specifically attempting to distinguish his "transposition" idea from Swedenborg's doctrine of correspondence in precisely the locations where he uses Swedenborg's term. For example, Lewis writes:

Where we tend to go wrong is in assuming that if there is to be a correspondence between two systems it must be a one for one correspondence—that A in the one system must be represented by a in the other, and so on. But the correspondence between emotion and sensation turns out not to be of that sort. And there never could be correspondence of that sort where the one system was really richer than the other. If the richer system is to be represented in the poorer at all, this can only be by giving each element in the poorer system more than one meaning. The transposition of the richer into the poorer must, so to speak, be algebraical, not arithmetical. If you are to translate from a language which has a large vocabulary into a language that has a small vocabulary, then you must be allowed to use several words in more than one sense. If you are to write a language with twenty-two vowel sounds in an alphabet with only five vowel characters then you must be allowed to give each of those five characters more than one value. If you are making a piano version of a piece originally scored for an orchestra, then the same piano notes which represent flutes in one passage must also represent violins in another. (emphasis added)

And a little later in the essay, along similar lines:

I am not going to maintain that what I call Transposition is the only possible mode whereby a poorer medium can respond to a richer: but I claim that it is very hard to imagine any other. It is therefore, at the very least, not improbable that Transposition occurs whenever the higher reproduces itself in the lower. Thus, to digress for a moment, it seems to me very likely that the real relation between mind and body is one of Transposition. We are certain that, in this life at any rate, thought is intimately connected with the brain. The theory that thought therefore is merely a movement in the brain is, in my opinion, nonsense; for if so, that theory itself would be merely a movement, an event among atoms, which may have speed and direction but of which it would be meaningless to use the words "true" or "false". We are driven then to some kind of correspondence. But if we assume a one-for-one correspondence this means that we have to attribute an almost unbelievable complexity and variety of events to the brain. But I submit that a one-for-one relation is probably quite unnecessary. All our examples suggest that the brain can respond—in a sense, adequately and exquisitely correspond—to the seemingly infinite variety of consciousness without providing one single physical modification for each single modification of consciousness. (emphasis added)

Lewis must here be critiquing some other line of thinking about this "transposition" or "correspondence." And given his liberal use of Swedenborg's term in these specific locations (he uses it in only one other place in the entire essay), and his likely knowledge of Swedenborg's doctrine of correspondence as a very precise relationship between spiritual and material things, Swedenborg's doctrine—and the Swedenborgians who hold to it—are Lewis's most likely whipping boy in his criticisms here of a "one-for-one correspondence."

It would go beyond the purview of the question to mount a full defense of Swedenborg's doctrine of correspondence from Lewis's apparent attack upon it here. Suffice it to say that Swedenborg was well aware that spiritual things can never be fully expressed on the material level for the simple reason that material reality is cruder, or in today's terminology, has a "lower resolution," than spiritual reality.

The relevant point for the purposes of the question is that Lewis seems here to be responding to Swedenborg's doctrine of correspondence. This suggests that Swedenborg's doctrine on that subject was indeed one of the precedents in Lewis's mind as he formulated his idea of transposition, and composed his essay on that subject.

Conclusion

Certainly Plato and the Neoplatonists were a major precedent of and influence on C.S. Lewis's theory of Transposition. This line of influence will have to remain for another person to trace in more detail.

In more recent Christian history, Emanuel Swedenborg's doctrine of correspondence is a clear precedent. Swedenborg did not use the term "transposition" to describe his doctrine. However, the parallelism with Lewis's concept of transposition is clear and compelling. And there is every reason to believe that Lewis was quite cognizant of Swedenborg's doctrine of correspondence as he formulated his thinking and penned his essay on "Transposition."