First off, I want to say this is a really good question that reflects real thought on an interpretative issue in Kant studies. Second, I think you're grasping some major things but also thinking backwards (by which I mean imposing contemporary categories on what Kant is doing).
In terms of your question, one major issue is going to be where in Kant you are reading. The account you are giving sounds closest to the Critique of Pure Reason (Kritik der reinen Vernunft). There, I take Kant's position to be as follows:
- As far as empirical science is concerned, our actions are determined
- Moral responsibility requires our actions to be free
This is what Kant calls the "Third Antinomy" (or perhaps that is just what contemporary Kant scholars call it).
But there are some specific details where I think you may be deviating rather sharply from Kant (or at least wording them in some significantly different ways). The background for this is that Kant's account of human knowledge is predicated on the belief that our knowledge is limited to objects (those things we place under the forms of sensibility and categories of the understanding). To put it another way, Kant is a skeptic about certain types of knowledge claims, because he thinks we never encounter things in an unencumbered way. But the manner in which we relate to objects is called "understanding."
It turns out that our actions are renderable both through our understanding which places things under the categories of cause and necessity (to name the most relevant two). But our actions take place for ourselves under the auspices of reason which is a different faculty. The easiest way to say this is that I understand your actions, but I reason to my actions on Kant's account.
So to return to the details in your question, you state:
Therefore, the reason must be part of the noumenal world, from where it is able to cause our behavior.
but at least on my reading of Kant, there's a minor error here. Viz., Kant does not see this as a conclusion but rather as a premise that relates to what reasoning is. But if you're just reading the Critique of Pure Reason as your source, then you might come up with this conclusion. The problem is that Kant changes how this works between the 1st Critique, the Groundwork, the 2nd critique, the Metaphysics of Moral, and Religion. Each of them gives a slightly different explanation of how we are morally/noumenally free yet empirically determined.
You're correct to think this progression is wrong:
We hold people responsible->We should hold people responsible->Holding people responsible would not make sense without free will (free from laws of physics/empirical influences)->There must be free will in the noumenal world.
No Kant work states this.
There's two important features in Kant's ethics that do seem similar. I take it that Kant asserts (at least outside the Groundwork this becomes a faktum of reason) we have reason (Vernunft) and this means that we can choose our own actions -- in fact, we can choose them despite our empirical natures. Thus, we are responsible for our actions, and thus, we are not determined.
A better but related argument has to do with Kant's belief that the world is ultimately just / what is called in the literature Kant's proportionality principle/thesis. Kant, for reasons that are nearly opaque, asserts in his moral philosophy that the world will ultimately turn out to be just. But this is contrary to empirical evidence about this world (evil people die happy; good people die sad; etc). Thus, Kant asserts a God brings about just desserts in an afterlife.
Depending on which work you are looking at in Kant's moral philosophy, the basis of the claim that we are free is different.
Critique of Pure Reason -> An antinomy shows that we cannot know we are free, but we live that we are free. Reason is free and distinct from understanding.
Groundwork -> Part III is an argument that we are somehow free (not well-accepted or understood)
Critique of [Pure] Practical Reason -> we are understood to be free as a fact of reason.
Metaphysics of Moral: Doctrine of Virtue -> we are free as a condition for a just world.
Religion within the Bounds of Reason Alone -> we are free at least insofar as we make a sort of root decision that impacts our later ability to be free.
Notably, it moves from an argument in the first two works to an assertion in the latter three.