A social construct is defined as an agreement between members of a society. — Harry Hindu
Not necessarily. Fear of a prolonged electricity outage could be considered a social construct, a wholly modern invention, while it is certainly not an "agreement" between anyone. It's outright undesired. But the intrinsic nature of losing something one desires, no matter how foolish and unnecessary, makes it widespread and common. Begrudgingly acknowledging what we do not wish to acknowledge is an agreement, but denial remains a true factor in society which invalidates any sort of conscious or willful agreement.
Gender as a social construct would be the agreement between members of a society on how each sex behaves. — Harry Hindu
This is where it gets weird. Fuzzy, if you will. I'll continue to your next sentence as to "expectation."
To identify as a one gender or the other would be identifying as an expectation society has of the sexes.
How is an expectation, or agreement among members of a society, an identity? — Harry Hindu
Expectation is fine. No one is forced to behave a certain way other than the basic codified laws. Don't murder people, don't park at Zone A after 6PM, etc., etc. ad infinitum.
Sure, people don't expect you to walk around in public cursing and grabbing one's genitals in front of mixed company, but legally, that's allowed. One might be socially ostracized, but as long as no codified law is broken (harassing, stalking, or assaulting said person) that's just nobodies concern.
To the point, an identity is what one holds as a fundamental core of their existence. For example, one might consider oneself a lover, another might consider themself a fighter, or an intellect, or a "blunt" person. That's their right (in most all countries) to pursue what they consider happiness, be it fulfilling or not, provided it doesn't break codified laws.
Parents often expect their children to be "good persons", and such might be considered one's identity. Don't you consider yourself a good person? That's one aspect of your identity sure. But, as I'm sure you agree, that doesn't cast an unchangeable mold as to who or what you are (or perhaps can be). Does it?
It is based on an understanding there are these biological realities of male and female a priori to the social construction and it is the social construction that is dependent upon these biological realities to exist. — Harry Hindu
This is what I would consider the key point of your argument. While I agree, surely plenty don't. And though I feel you've answered this quite succinctly, surely
@Michael will address this quote specifically.
Society is not saying that wearing a dress makes you a woman. Society is saying that you are a female and we expect you to behave this way because you are a female. — Harry Hindu
Society can say whatever it wants. I mean, sure, in theory, society can pass a law saying people under 5' should be killed for their own well-being, or perhaps that certain people should be enslaved. It doesn't make it right. Naturally, one must adapt to survive. So even in unjust times and scenarios, one would be wise to, I suppose, "get with the program" and do what you must do to survive. This has been the one consistent reality since the beginning of time. But, that doesn't change the underlying deeper reality that just because something is how it is, doesn't mean it's how it should be or would be best for all. You can agree on that much, yes?
In short, social expectations don't conflate with modern legal systems that separate persons by sex during moments of vulnerability (using the bathroom, being seated for a prolonged period with one's pants down and around their ankles thus immobilizing the person). Men have a primal desire to mate (engage in intercourse) with women. Anyone who avoids that fact is simply ignorant of the larger discussion. And human persons born a male are liable to retain such desires regardless of artificial medical operation. Am I wrong?