If a married Jewish man undergoes the hormonal and surgical procedure of sex change, and "he's become a woman".
Would he still have the Halachic status of a man, at least relating to his marital status?
Could he give his wife a Get if he wanted to?
Would he be deemed by Halachah as mentally competent to give the Get?
According to the majority of poskim, he is a man despite whatever surgery he had and he would certainly be allowed to give his wife a get. According to the minority view of the Tzitz Eliezer, he would not have to give his wife a get, because the marriage dissolved when he "became a woman."
He obviously does not meet the qualifications for a Shoteh. See R. Bleich, Contemporary Halakhic Problems, vol. 1, p. 100ff.
The Tzitz Eliezer is at the end of Vol 10 25:26:6. The teshuvah discusses transplants, and he raises the question posed here as a hypothetical. A second teshuvah, in vol 11 #78, is about someone with a hormonal issue impacting their sexual development during gestation who is born looking female, but has one testicle. In both cases he says that gender is decided by external organs, not internal ones.
I frankly do not know of anyone else who wrote on the topic. Also, notably, REW wrote a teshuvah, i.e. intending to be follows in practice -- lehalakhah ulema'aseh, as opposed to R Bleich's more theoretical journal article.
Speaking theoretically, I could see logic of a third position.... That with regard to halachic gender differences based on role, the person's gender is that of their birth. So that the person in question would still be obligated to wear tefillin, daven three times a day, etc... But when it comes to gender issues related to sexuality, their gender is their new assigned gender. After all, if the person is now female, is yichud with men or with women more problematic? What to do about minyan, which is interpersonal but not sexual, is beyond the scope of my speculation.
I heard that the London Beis Din sometime in the 20th century had such a case and asked Rabbi Ruderman if this person could give a get. Rabbi Ruderman didn't discuss if the gender change would have any ramifications on their halachic gender. He responded that even if they are now considered a female, it doesn't say anywhere a woman can't give their wife a get. The Torah says give your wife a get, and they did.
Whether they are a shotah or not would depend on the case. The case you describe doesn't sound like they're so crazy to me...
