The most glaring problem, here, in my eyes at least, is a "bad company" objection. Premise (3) is meant to be the kicker, in that (though this is not explicitly stated) an infinite regress of choices is supposedly impossible. However, if we accepted the idea, we might wonder:
- Suppose that every effect has a cause, and every cause is an effect of another cause, and so on. So, for there to be any effects, every effect must be preceded by infinitely many causes.
- In normal epistemic logic, for kS, k = "it is know that" and S = some sentence, it is also true that kS → kkS, and so on down the line.
- Not only the k-operator, but a proof operator, can face such a regress. Indeed, this is where rejections of foundationalism or advocacy of infinitism come into play. Another similar case would be for an understanding logic, such that uS → uuS.
- Modal logic can encode such a regress, though the prevailing assumption is that iterated modal operators are trivially reducible such that ◊◊S → ◊S, for instance.
So let's consider (4), here. Strong free will is the kind that depends on alternative possibilities, so it would have to connect up with modal logic. If singular possibility encodes for infinite possibility, and this unobjectionably enough, why would strong free will not work similarly? You are effectively claiming a decision operator such that dS → ddS, etc. I don't see a problem with a reversed collapse as such, from infinitely many decisions to one decision.
Another problem, though, is the idea that if ddS is involuntary, then so is dS. Inasmuch as we do not make choices in a vacuum, it is safe enough to say that the very fact that we face a decision problem can be caused by the historical environment (though per Kant (see below) this claim that transcendental freedom's "presence" is discernible in the physical world, might be rejected). A fact with a conditional structure, whose metaphysical antecedent is one subfact, but whose consequent is a disjunction of possible future subfacts, seems to be the kind of fact "inside of which" free will (or choice, rather) would occur. In other words, ddS (or dddS, etc., for that matter) might well be strictly determined by what came before, and yet dS itself would still be disjoint in effect, hence "free."
It is a mistake to think that belief in the possibility of free choices, or even the actuality of a faculty with such power, is necessarily a belief that every act that an agent with said power does, is a result of free will/choice. Kant thought that, per our phenomenal character, it would be possible, with an advanced enough science, to know all the empirically strict causes of our tangible behavior; but for all that, he located transcendental freedom and its causal spontaneity, then, in the noumenal realm. Sometimes people think this means that Kant is suggesting that our noumenal choice involves manifesting, or at least entering into, specific possible worlds, as though our souls could survey these worlds "from on high," and the implicit freedom of our empirical actions is the freedom to choose which set of empirical actions to incarnate in. That's fairly exotic a claim, and Kant might have looked for other abstract descriptions of how this procedure is actually supposed to work; but as far as his bold (and paradoxically, but epistemologically, modest) claims in his known work go, he openly admitted that he did not know, and could not know, much less more deeply understand, how transcendental freedom selects from the empirical possibilities, such as to allow us to be noumenally independent of physical causation and yet, in the phenomenal realm, predictably subject to the same.
More pointedly, though, right now, I would ask: why believe that either all our actions are determined or all are freely willed (in the strong way)? And mightn't some of our actions be random, no less, too?
So I don't know that your argument is unsound in the normal sense of that term, but it doesn't seem sound, either. Abstract, and obscure, philosophical premises are hard to judge on this kind of score.