The Argument of Reason is a bit complex - I think much of your confusion is my fault, since I’m probably not explaining it very well.
One thing to point out is that the argument is not attempting to prove the existence of the soul in a Christian sense. I think it would be a misunderstanding to say that the argument is conflating the mind with the soul. The argument is simply trying to show that one aspect of our minds - namely, our ability to reason - cannot be explained using a Naturalistic system. If the argument is successful, it will show that Naturalism is fundamentally unable to account for human reason. This leaves us with two alternatives.
- Naturalism is true, and human reason is an illusion
or
- Human reason is objectively real, thus, Naturalism should be rejected in favor of some other worldview, like Supernaturalism, Mind-Body Dualism, or Idealism.
Since I believe that 1 (namely, that reason is illusory) leads to a self-contradiction, I cannot view it as being true. Thus, the only option left to us is 2. (The argument can then be further developed to show that reason is the result of a supernatural, eternal, ‘mind’ for whom reason is a necessary component. However, at this point I think we’re getting ahead of ourselves.)
You pointed out that certain mental diseases - like Dissociative Identity Disorder, give us reason to believe that the mind is entirely physical. However, I think this would only work if the Argument from Reason sought to show that
personality cannot be material. In reality, the argument’s goal is much more modest - that only our ability to reason cannot be accounted for by natural processes. While I do think such mental disorders raise interesting philosophical questions, particularly in regards to the formation of personality, I don’t think that it serves to undermine the argument.
Again, the argument is not chiefly concerned with souls - at least not in the Christian sense of the word. I do, of course, believe that the soul exists, and I do believe that it is related to the mind, though I’m not sure if I’d conflate the two. But I think that the argument gives good reason to believe that materialism is false and that some sort of mind-body dualism is true. If reason exists, and if it can’t be accounted for by naturalism (in this case, bodily processes) then I think that we must conclude that at minimum, one specific part of our mind (our rationality) is non-material. Since Naturalism, by definition, does not allow for the existence of anything outside of ‘nature’ then I think we have to conclude that it is false.