I am not trying to continue this interchange; but, since you asked some questions, I will respond.

"What do you mean by objective - do you mean based in some physical reality, such as biology, or a region of the brain?"

Objective means real, existing in and of itself regardless of perception. Reality may be physical or non-physical, depending on your cosmology. If you're a materialist, then that's not true; but I have yet to meet a pure materialist, since pure materialism necessarily leads to the conclusion that personhood is fiction and that man is nothing but a machine made of meat. People show what they really "believe" by the way they live and the choices they make; and, whatever people may say about believing in materialism, no one seems to actually believe that.

"Because the implication of that is that gender roles and all the sexism and discrimination and oppression that comes with the idea that gender is objective, are ‘natural’."

There is no such implication. You are conflating gender, the objective reality, with gender as it is subjectively perceived. You don't have to deny the objective reality of gender to avoid this sort of prejudice. The key is humility: the recognition that our perception is imperfect, and that there is a difference between the objective reality that is gender and our perception of it.

Look, it's not your fault you're not clear on this. Virtually everyone is making this mistake. It's subtle.

"This is the biological essentialism that justifies sexism, patriarchy, and transphobia."

Even if gender is a purely biological phenomenon, the implication you mention is not necessary. Again, the difference is humility: the recognition that we do not understand this phenomenon perfectly and never will, therefore we must remain open-minded. That's science.

Do you realize you are making the same mistake people like Matt Walsh and Michael Knowles are making? Only you're making it on the other side. Both of you are conflating fact and belief, objective reality and subjective perception. Knowles believes reality is his perception, while the logic of your position is that perception is the only reality. You may deny this, but you have demonstrated it here, in your reply. You have explicitly tied believing that gender is essential to the list of evils you mentioned, as a matter of necessity.

"If gender is based on something objective, then why do gender roles vary across time and culture? Why are there different cultures with vastly different gender norms now? Why are we able to show that gender norms, that gender roles, that the way we understand and express gender has varied, measurably, within our own lifetimes?"

Because our perception of gender changes, not gender itself.

"All of this implies that gender is not objective, but socially constructed, and malleable - as well as real, meaningful and consequential."

No, it means the perception of gender is not objective. It is that perception that is constructed.

"Gender is our socially constructed interpretation of that anatomy"

This cannot be true. It is true of our perception of gender, not true of gender itself.

Failure to appreciate this distinction has led many trans people to grief, and it will always, eventually, lead to despair.

I think, by this point, I may have lost you; so let me illustrate with something obvious.

Some people used to believe that the world was flat, while other people believed that the world was round; but both sides admitted that the world existed to begin with. Those who believed the world was flat might have made laws forbidding crazy sea captains from taking crews beyond the horizon, putting their lives at risk. That's kind of what the Right is doing now with gender. The correct approach is to let people decide for themselves whether the world is flat or round, and let them sign a waiver so that crazy sea captains can crew their ships and sail where they want to. You don't have to deny that the world is "essential" to do that.

Conflation of subjective perception with objective reality is the logical flaw at the foundation of this misunderstanding.

"Actually, it’s very similar to people who say ‘I don’t understand transgender people, so they aren’t valid’. You’re essentially saying ‘I don’t understand social construction, so it isn’t valid’."

You astonish me. Aren't you saying the same thing? Aren't you telling me that I don't understand social construction, so my view isn't valid?

If we aren't allowed to say that we think an idea is incorrect, that is the same thing as saying that all our understandings are perfect and there is nothing new to learn.