I find four questions in your question. They are linked, so that is not complaint.
- "Does gender identity require a dualist account of consciousness"?
NO. It requires a social perspective on sex. As you say, sex is used, in this context, to mean something biological, physical. But sex is built in to social forms in all sorts of ways, and so, by distinguishing the biological functions from the social dimension of sex, it is possible to articulate the concept of gender identity as a social role. A complication here is that the distinction between the biological and the social contested, so that issue is more difficult than it might seem.
2."Does this necessarily mean that another, private entity must be responsible for such self-knowledge, aka the mind...?"
It depends what you mean by "private" and "entity". It is obviously possible for people to have beliefs and thoughts that they do not tell other people. That doesn't amount to an entity. But clearly it is possible for people considering gender identity to tell other people, so it isn't private in the sense required by Dualist theories. So the answer to this question is, No.
Many philosophers think that privacy postulated by Dualist theories is incoherent. In that case, the answer to your question is No. For more about this See Stanford Encyclopedia of Philosophy - Private Language
- "Surely such an identity cannot be explained by physical processes?"
There don't seem to be agreed explanations of any kind available at present, but explanations by physical processes may well become available one day. In the mean time, explanations by psychological and social processes are available and might well prove helpful.
I should point out at this point that since gender is a social role, it needs no more, and no different, explanation that the choice to get married or become a philosopher.
- "Or, is this guilty of making what Ryle objected to as the 'Category Mistake'?"
It's not clear what "this" refers to. If you mean "another, private entity" then the answer is Yes. See Stanford Encyclopedia of Philosophy - Categories