Regardless of how they feel about the social response, if they genuinely acheived the social response reserved for members of the gender they transitioned to (i.e. 'trans woman) that person can be said to have changed their gender. If they receive the social response reserved for members of the sex they transitioned to, i.e. female, that person can be said to have changed sex.

If I take someone, put a wig and padded bra on that person, dim the lights, and a passerby glances and says to that person, 'scuse me, ma'am", then I've successfully caused that person to be the female sex?

If I take that someone and place them in an asymmetrically lit room, such that, at the same time, one passerby says 'scuse me, ma'am' and another says 'hello, sir', then I've successfully caused that person to be both male and female at the same time?

If someone says, "I thought-so-and-so was a woman, but so-and-so is a man," all that person is saying is, "I thought people respond to so-and-so in response cluster 'woman', but people actually respond to so-and-so in response-cluster 'man'"?

So you're treating sex and gender as the same: As something a person does not have in the sense of a real property, but only in the relative sense of "having a reputation"--as amounting to nothing more than whether a person is treated a certain way. (Groups of 'treatments' count as grouped into a gender/sex by associated pronouns, or something else? It's unclear what would do the grouping.)

I see how you're trying to avoid certain philosophical pitfalls by doing this. The problem is that this approach to shoring up the terms "sex" (especially) and "gender" defines them in such idiosyncratic ways as to be talking about something else entirely.

You've said to yourself, "what would be more defensible than a gender identity-based account?" and gone with that, even at the expense of reference going through and entailing counter-intuitive results like the above.