I hadn't actually questioned this before. I knew that it started out as a joke, but then when Harry actually embraced the role of Harriet, I was surprised that I hadn't seen this coming. Of course Harry was not good at being a boy, and since being Harriet made her actually be happy, of course she thought of herself as female, then.
But is Harry/Harriet actually a woman?
I'm not sure, as yet.
I actually get the feeling that Harry/Harriet would be parasexual (not asexual, but something transcending mere male and female) if society allowed. It doesn't, really, but female's a more comfortable mask to hide her tender spirit better, and Harriet can be a happy girl, a popular girl, with the same traits that made Harry an unpopular and bullied boy (and just a little bit of coaching).
Harry/Harriet's problem is that Harry's mother, instead of helping Harry cultivate a social mask that didn't chafe too much and choose his battles carefully, decided that there was no problem with him going about sharing everything he held most precious and dear with the entire world with no correction on anything other than items of fact. She would correct him if he spelled something wrong or wore shorts and sandals out in the snow, but not if he picked his nose in public.
When he didn't make the connection between "I smell funny" and "I get beat up on", she didn't help him make it. She went railing off about individuality and how society does not accept the true individual, and never helped him find the little ways to hide and compromise that most people who can function in society wind up finding. It would be great if we could all go about naked, mentally and physically. It would be wonderful. But we can't, so we hide those parts of us that can't bear to be touched. No one taught Harry about this, so he grew up inside-out.
By the time he hit his teenage years, I'm not sure anyone could have taught him how to be a socially acceptable young man, but Sally (a girl his age, the little sister of our heroine) saw that she could teach him how to act like a socially acceptable -- even popular -- girl. He accepted the challenge on a dare, but grew to identify with the role, and to a large degree, with the gender.
I think Harriet will continue to act, and almost be, female, at least until s/he reaches middle age, at which point s/he will have developed enough of a sense of self to uncover which, if any, gender s/he actually is. I don't think that her current identification with the gender is indicative of her true nature; I think it's more along the lines of a drowning person clinging to any possible source of rescue.
I think Harriet is somewhat more transsexual than Dono Vorrutyer.
Dono made the change because it was politically advantageous, and because one of Dono's defining traits is a need for/love of power. Donna/Dono will not want to be a woman on Barrayar unless/until being a woman on Barrayar has as much capacity for overt, socially-accepted power as being a man has. (On Beta Colony, it doesn't matter.) I think Donna would have made the transition to Dono with 20th Century Earth medicine if doing so gave her male status in the current Barrayaran gender-sociopolitical climate.
I think Harriet would not make the medical transition from male to female with current Earth medicine; Harriet would be more than happy to make the medical transition with Betan medicine. However, Harriet would not need to make the transition with Betan therapy and Betan gender-role ambiguity. Harriet wants -- needs -- social acceptance without compromising too much of who s/he is. And right now, that means looking like and acting like a woman.