Formulate an argument for the case of philosophical determinism (i.e. no free will) that cannot be solved by the introduction of any psychics-free metaphysical entity.
First, determinism is not the same as the assumption that free will does not exist. Nor does even one entail the other. There are hard indeterminists, who believe in indeterminism but not in free will and compatibilist, who believe in determinism and free will.
I also understand "solved" in your question as "contradicted".
Now, determinism is, in most cases, conceived as the idea that by fixed natural laws an earlier state of the universe, even as early as its beginning, determines any later state.
A well-known illustration of this view is Laplace's demon, an enormous intellect, who can perfectly predict the future, not by supernatural means, but by knowing the speed and position of every atom in detail and calculating (like an unimaginable super computer) in advance how the universe will develop.
Yet, we can understand determinism differently, namely that the future is fixed. The B-theory of time claims that the passage of time is an illusion: past and future moments are not less real than the present. While the B-theory does not entail that there are rules by which past states determine future states, nonetheless there still is no way any event would have been able to turn out differently than it is observed.
If determinism is understood in this sense, it cannot be contradicted by metaphysical entities, which usually refute determinism, like a source of true randomness.
Now, regarding to entities like souls:
Of course if the soul exists and is free in the sense that its existence decisively establishes libertarian free will, it must refute determinism and nonexistence of free will by definition. There is no way around it by the most basic logic.
But we may understand "free" just in the sense that the soul or consciousness is independent from physical laws, i.e. it is conceived dualistically with mental-physical causation. And indeed, if we think about it, such entities may not be sufficient to refute determinism. Because there may still be psychological laws, by which the soul/consciousness operates and which may be deterministic, too.