There is so much confusion about "gender." It's a term everyone uses and no one understands.
Gender cannot simply be a category or label we use to collect a bunch of artificial qualities. It cannot be a "social construct." To say that gender is these things is to say that it has no objective existence -- that it isn't real, separate and apart from whatever we think about it.
Why is this a problem? Because if gender isn't real, then being transgender isn't real. And if being transgender isn't real, i.e. is an artificial category / social construct, then those of us who are calling ourselves transgender are deluded, or worse. This is an intolerable result; therefore, gender must have objective existence.
So, where is it? Maybe it's in the brain and we haven't found it yet; or maybe it's not a material quality at all. Maybe it's metaphysical, like the soul. But, either way, it's currently intangible -- which is not the same thing as "unreal." It just means, we can't prove it. It also means, our antagonists can't disprove it.
This is the only way to look at gender, if you believe being transgender is something real.
I think our community must come together on this, and soon, because the Right is gaining momentum; and, if we aren't clear about where we stand, they'll take control of the language and thus the conversation.