To repeat the question raised in the OP:
Now, if this were true, it would mean God cannot exist, because He
would be metaphysical. Most theists accept that God created the world,
which would be a non-physical cause for a physical concept. How do
theists reply to this, given that strong arguments exist for the idea
of causal closure (at least, that's what I've seen)?
According to Alvin Plantinga in Where the Conflict Really Lies (WTCRL), some theologians (he mentions in particular Rudolph Bultmann, John Macquarrie and Langdon Gilkey (WTCRL, pp 69-75)) accept the causal closure of physics to such an extent that they deny the existence of miracles although they maintain that God still created and sustains the universe. He calls this "hands-off theology" (WTCRL, page 72):
According to Bultmann, a divinely caused miracle or any other special
divine action would constitute God's "interfering" in the world; and
that, he says, can't happen. Bultmann's idea, shared by the others I
mentioned and many others, is what we might call "hand-off theology":
God creates the world and upholds it, but for the rest can't or at
least doesn't act in it; he steps aside and lets it evolve according
to the laws he has set for it.
This is not Plantinga's view.
He approaches the causal closure of physics by noting that even Newton "didn't accept hands-off theology" (WTCRL, page 77). Underlying causal closure is that it applies only to isolated and closed systems: "In classical physics, the great conservation laws are stated for closed and isolated systems." (WTCRL, page 78):
These principles, therefore, apply to isolated or closed systems.
If so, however, there is nothing in them to prevent God from changing
the velocity or direction of a particle. If he did so, obviously,
energy would not be conserved in the system in question; but equally
obviously, that system would not be closed, in which case the
principle of conservation of energy would not apply to it. Indeed,
there is nothing here to prevent God from miraculously parting the Red
Sea, or changing water into wine, or bringing someone back to life,
or, for that matter, creating ex nihilo a full-grown horse in the
middle of Times Square.
What this shows is that the causal closure of physics is not adequate to get the conclusions of hands-off theology. One also needs determinism. (WTCRL, page 85)
What, exactly, must be added to the Newtonian picture to get the
Laplacean picture? Determinism plus the causal close of the physical
universe. Although this addition is not at all implied by the physics
(as I said, it's a philosophical or theological assumption), it was
and is widely accepted, and indeed so widely accepted that it is often
completely overlooked in contexts where it is crucial.
However, this old determinism on which hands-off theology rests breaks down with the indeterminism introduced by quantum physics. At this point Plantinga dismisses hands-off theology (WTCRL, page 96):
Rudolph Bultmann, Langdon Gilkey, John Macquarrie, and their friends
rejected divine intervention in the name of an eighteenth-century
picture of science; many contemporary writers on religion and science
also reject divine intervention--not, now, by appealing to out-moded
science, but for other more obscure reasons.
To return to the original question, although some theologians based on Laplacean determinism and the causal closure of physics maintained a hands-off theology where God does not interfere even they did not assume that God did not exist because of this. With quantum physics one can today dismiss their hands-off theology as out-moded-science theology.
Plantinga, A. (2011). Where the conflict really lies: Science, religion, and naturalism. OUP USA.