The premise of your question is unjustified. It is a very widespread view that Finnegans Wake is precisely a postmodernist work. Take a look, for instance, at this answer on here, which states:
Perhaps the earliest great postmodern author was James Joyce. His
novel Finnegan's Wake [sic], for example, cannot be read as a typical
narrative as it makes little sense. It is, instead, a multi-layered
piece of work relying on puns, polyphony and an extreme stream of
consciousness technique to create a literary landscape that the reader
is free to interpret as they choose. Ulysses is less extreme, but can
also be considered postmodern.
A little googling will reveal many similar examples.
Matt Thrower already has a much better answer, which I encourage you to upvote. This answer, however, has received criticism for not using an authoritative reference and for suggesting that “a little googling” might help. My intention was merely to point out that many people do indeed consider Finnegans Wake to be postmodernist, and so the premise of the question is wrong. Since this is a discussion on StackExchange it seemed appropriate to refer to an answer from that forum.
The comments did make me think about who would qualify as being an authoritative source on this. As an author of Postmodernist Fiction (1987), Constructing Postmodernism (1992), Introduction to Postmodernism (2015), and (with others, ed.) The Cambridge Companion to Thomas Pynchon (2012), I believe Brian McHale fits the bill.
In Postmodernist Fiction, McHale discusses Finnegans Wake at length, contrasting it with Ulysses. At the end of the book he compares the endings of four of Joyce’s prose fictions and writes:
In “The Dead” and Ulysses, the simulation of death has been passed
through the medium of an individual consciousness, “an ordinary mind
on an ordinary day”—Gabriel Conroy’s mind, Molly Bloom’s mind. These
texts are, in the first place, representations of minds, and only
secondarily representations of the onset of sleep and, by extension,
of death. The formal technique is, in one case, free indirect
discourse (“The Dead”), in the other direct interior monologue
(Ulysses). But in both cases, it is through the represented
consciousness of the character that the represented world—whether
immediately present, remembered, or anticipated—is filtered to us. And
this world is stable and reconstructable, forming an ontologically
unproblematic backdrop against which the movements of the characters’
minds may be displayed.
Modernist fiction, in short.
The end of
Finnegans Wake, too, represents an interior discourse, that of Anna Li
via Plurabelle. But hers is not a consciousness like Gabriel Conroy’s
or Molly Bloom’s, not “an ordinary mind on an ordinary day,” but more
like a collective consciousness—“Allgearls is wea”—or even the
collective unconscious located in language itself. Molly Bloom’s
soliloquy notoriously represents the “stream of consciousness,” but
Anna Livia is the thing itself: the personification of the River
Liffey, she literalizes the metaphor “stream of consciousness.” Just
as her discourse seems to sweep up all language in its stream, so it
also sweeps up the projected world of this text: there is no stable
world behind this consciousness, but only a flux of discourse in which
fragments of different, incompatible realities flicker into existence
and out of existence again, overwhelmed by the competing reality of
Postmodernist fiction, in short.
Taken together with McHale’s other arguments, this is perhaps sufficient to show that Joyce’s writing in Finnegans Wake can reasonably be seen as postmodernist, and is indeed considered as such by prominent writers on the subject.