But to say that gender is socially constructed, and thus that social construction plays a role in gender identity development, isn’t at all to say that gender isn’t real.

I think it is necessary to distinguish clearly between reality (objective) and perception (subjective). Labels are subjective and artificial -- constructed -- by their very nature; the key question is whether they correspond to something objectively real.

The purpose of labeling gender a "social construct" can only be to delegitimize its objective nature. If you're trying to say that gender is a label for something real that we do not perfectly understand, then say that; but simply calling it a "construct" of any kind is inherently dismissive. This is because it's redundant; and there is no point in being redundant unless you are trying to emphasize the quality to which you are referring, here the "constructed" nature of the concept.

You may not intend to do this when you call gender a "social construct"; but that's what laypeople hear -- and I suspect it's the dirty little secret underlying even some professionals' use of the term.

There is a tendency, I think, among both laypeople and professionals of one sort or another to make war on traditional notions of gender; and their reasons for doing so aren't always disinterested. Smashing existing paradigms can be a means to an ignoble end, e.g., having permission in one's own mind to call oneself whatever the hell one wants. The reasoning is, "Well, gender is nothing but a social construct, anyway, so who's to say I'm wrong? My calling myself moon-gender is just as legitimate as you calling yourself female." This is the way the "social construct" frame comes across to many people: that gender does not correspond to something limited by objective reality, and is instead infinitely malleable according to the wishes of the individual. Again, this may not be the intention (though I think it is, surprisingly often), but it's the inevitable result of this frame.

A right-wing reactionary might say that, because gender is socially constructed then being transgender isn’t real - but they would be doing so based on a fundamental misunderstanding of what gender and social construction are.

I hope you aren't calling me a right-wing reactionary. I assure you, I have given a great deal of thought to this question, because it has led to no small amount of personal pain among transgender people. I appreciate that my grasp of gender may not square with the way the label is defined by sociologists; but truth is not a matter of fiat in science, and my experience has been that those who speak of gender being a "social construct" are being self-contradictory without realizing it. They simply haven't thought through the necessary implications of their frame.

I wouldn't mind that much if I hadn't seen how much people have struggled with gender as a result of this mischaracterization.

I don't want to argue, so I'll end there.