There is even more. As I noted in an earlier post, in his moral theory, Kant postulates a noumenal self as a moral agent. These two selves are in addition to Kant's empirical self, which is how I ordinarily, pre-philosophically appear to myself as a person. This is Kant's famous theory of the three selves, and all three look real. Once again, one must not confuse materialism with realism.
Selves can be real.
So, much like Wittgenstein, Kant has lots of realist aspects. Now I come back to your denial that Kant's arguments are not psychological claims about the workings of the mind. To begin with an argument is never a claim. Only a premiss or conclusion of an argument can state a claim. I stated above is precisely a psychological claim about the workings of the mind.