If gender is a social construct, how can there be a neurological mismatch between that and their biological sex?
At the risk of vastly oversimplifying a very complex topic:
The biological component refers to the elements we normally oversimplify as "male" and "female," which usually means phenotypes such as chromosomes, genitals, breast development, and hormone levels.
Gender, as a social construct, is how we as a society make sense of these biological features (typically, only the most obvious ones such as genitals).
However, it's not like there is a line between the two that cannot be crossed. Social structures can actually modify your brain; what you learn, and how you approach the world, is often encoded in neurological structures, and this has all sorts of cognitive impacts. One example is that the ancient Greeks didn't have a word for "blue." Their eyes worked the same as ours, but because they did not distinguish "blue" as a specific color, they simply didn't perceive it that way. Homer, for example, never used the term; he described the oceans as "wine-dark." Contemporary members of the Himba tribes also do not perceive blue as a separate color -- but they can also perceive distinctions in shades of green that we do not.
(It runs in the other direction, too. E.g. stress causes all sorts of physiological problems, like high blood pressure, because the emotional experience causes the body to release hormones.)
We also know, in part due to research in transgenderism, that children internalize gender norms at a very young age. A biological boy who identifies persistently as a girl knows many of the social cues we typically associate with girls -- e.g. she will play with dolls, want to wear dresses, prefers pink and so on. Our society has determined that "pink is for girls" and "blue is for boys," and thus that is how, from even a very young age, we make sense of our bodies, what our bodies tell us about ourselves, and what our bodies tell others about us.
Thus, what is probably happening is that there is a neurological condition which convinces the individual that, even if they have a fully functional penis and testes, that they are "not a male." The result is distress, often extreme distress, that cannot be resolved by therapy or psychiatric medications.
The (fairly rare) presence of persistent gender dysmorphia in children, carrying through to adulthood, is a strong indicator that this is not a result of some sort of event, or trauma, or psychological pressure. Other hints include subcultures like drag, where the individual does not experience distress because of an underlying mismatch between gender and biological sex -- rather, they are deliberately performing as a member of the opposite gender for entertainment, pleasure, amusement, or social commentary.
It may be very similar to the body integrity identity disorder mentioned earlier in this thread, where the afflicted individual has a neurological condition that convinces the patient that a 100% healthy limb is somehow "wrong," or is not really a part of their body. It is not a psychological issue, the individual is not delusional, they can't be talked out of it, psychiatric medications and therapy rarely work.
Perhaps at some future date, we might be able to deal with BIID and transgenderism in the same way, by addressing the neurological issues. But, maybe not. Even if that is possible, the only way to find develop those treatments is to acknowledge and discover the physical basis for the mismatch between one's biological features and one's gender identity. Either way, at this time, the best option is not to deny how the individual feels, or force them to match their gender identity to their biological features.
Perhaps society needs to be okay with effeminate men and masculine women, because trans ideology seems to be re-enforcing gender stereotypes.
Yeah, no.
Transgenderism isn't about "effeminate men" or "masculine women." There are plenty of men who enjoy things that we stereotype as "female" who don't feel psychological distress because they have a penis; and the same for women. Those very labels presume that gender is purely or primarily biological in origin.
Society
should recognize that gender norms are flexible social constructs, and let people be who they want to be. That doesn't mean we should completely ignore biology, including how the vast majority of individuals are happily cis. We should recognize that gender norms can be toxic and oppressive, and fuel misogyny, sexism, homophobia and transphobia.
Transgenderism threatens, rather than supports, the standard narrative of cis heteronormativity. It is a direct assault on the flawed assumption that our gender norms are based on iron-clad biological features. That is one key reason why there is so much transphobia.