Gibberty:
This reply is kinda long, but it's highly contentious to claim that sex is a social construct, and then to further argue that because sex is a social construct males can change into females and vice versa. The first claim is false, and the second claim just doesn't follow.
What is a social construct? Money is a good example of a social construct. Humans have got together, as a society, and decided that certain objects (flat metal discs, pieces of paper) constitute 'money', and that money has a certain value. One can exchange money for other goods. One way you can tell if something is a social construct is to ask yourself a question - if all the humans suddenly disappeared, would that thing still exist? Or does it need human society to continue to exist? Money is clearly a social construct. Without human beings, money is just bits of paper with no value. Without society supporting the construct, money quickly becomes useless and meaningless.
Sex is not a social construct. In philosophy, for instance, philosophers acknowledge that sex is what's called a 'natural kind' - it's a stable cluster of properties that are objective, and real in nature (other natural kinds include species, water, gasses, plants, etc). Sex is not a social construct because if all the human beings disappeared sex would still exist - animals and plants have sexes, and reproduce sexually. Even if society broke down, sex would still exist. We would still have males, we would still have females, and reproduction would still occur. Sex exists no matter how humans think about it. Sex isn't 'assigned'. Rather, it is observed - there are certain physical properties (genitals, for instance) that are highly reliable indicators of one's biological sex. A mother observes her newborn has a penis and infers, quite rightly, that he is a male.
What IS a social construct is the meaning we attach to sex (which was traditionally called 'gender'). It's a social construct that the female sex are weak,hysterical, stupid, emotional, fond of fashion, etc. So the significance and importance we attach to sex may very well be a social construct, but sex itself is not. So your first point is wrong.
Suppose we grant that you are actually correct, though, and that sex is a social construct. Does it thereby follow that males can change into females and vice versa? Does this render the social construct arbitrary and fluid? No, it does not. To return to the example of money; money is a social construct but that doesn't mean I can declare that my 10 pound note is actually worth 100 pounds. Once we agree on what social constructs mean, and we set the boundaries and parameters, the social constructs end up with truth conditions attached to them - they can't just be arbitrarily changed by the whims of individuals. We have all collectively agreed what the value of money is, and I can't just unilaterally declare that it means something else.
To return to sex, even if sex is a social construct, we've still decided as a society that sex cannot be changed once it is 'assigned' (to use your terminology). It is not part of the social construct that males can become females, and females can become males.
Note, you are also contradicting yourself by saying that sex is a social construct, and yet the assignment turns out to be wrong in some individuals. If sex is purely a social construct, then the assignment of sex at birth settles the matter of what one's sex is. That's just what sex is, on the social construction view - an assignment at birth. When you say that the assignment 'turns out to be wrong', you are implicitly relying on there being something besides the act of assignment that constitutes sex. What you're really saying is "in the case of certain individuals, the assignment gets their sex wrong because their REAL sex is actually female." But on the social constructivist view there is no such thing as REAL sex independent of the assignment. So you're stuck. You have to either accept 1) sex is a social construct and assigned at birth, which means that your assigned sex just is your sex - period. Or 2) there is such a thing as sex, objectively speaking, that can clash with an 'assigned' sex at birth. And this is what someone's real sex is. But that's not a social constructivist position - that's a realist position about sex. Hope this clears things up.