OK, several questions. Some of these ideas I did in fact cover in an earlier version, but cut out for space, so happy to revisit here.
Let’s take this first: How can you be the wrong gender if gender is a social construct?
Describing gender as a “social construct” doesn’t make gender any less real. Most of the things we live by are social constructs; family relationships, laws, prejudice, borders, money… everything that comprises our society is a social construct. All of those things affect your life in profound ways.
Even mundane things like traffic signs are part of a social convention, but there are real consequences to running a red light, up to and including death. Those social conventions are what we live by, minute by minute in every aspect of our lives.
This idea that gender is meaningless because it is “social” is a red herring. It’s one of the ways privators try to prise apart gender and biology, to advocate that biology has priority. But the social is the primary thing we live by.
Next: How does a trans person know that their gender was misassigned?
How does anyone know what their gender is? Cis people just know what their gender is, right? Same with trans people.
Asking this is liking asking someone how they know what their sexuality is. They don’t have to give you a justification for how they know — they just know.
Next: How does a trans woman know [she] is not a gay man but is, instead, a woman?
There’s a major misconception here that being trans is somehow intertwined with sexuality, which is incorrect. Trans people can be gay or straight, any sexuality; being trans is simply orthogonal to that. In essence this is the same as the previous question, but with a poor choice of pronoun thrown in.
Finally: If gender is not essential, isn’t it fair to ask whether feeling trans is a perceptual dysfunction?
On a similar basis you could, but wouldn’t, say: “If money is a social construct, isn’t it fair to ask whether feeling poverty is a perceptual dysfunction?”. The idea doesn’t hang together because, as above, social constructs are real: they are the basis of our experiences.
Trans people benefit from transition: that is the key thing. Arguing about whether they are “correct” in how they feel is irrelevant.
And we can also just point to the medical consensus here.