Trans is a social construct, largely because everything is, even science. Since trans is both a social construct and a people grouping label, it suffers from the usual vulnerabilities. How do you decide who is in the group and who isn't? How do you decide who represents the group?
I don't think those who consider themselves trans have the luxury of being able to determine who gets into their group. Anyone can just say they are in the group, even if the current members of the group would rather not have them.
Worse, those who group themselves as trans don't have the luxury of being represented by legitimate voices. There are no democratic votes to determine, for example, who will stand for the trans and speak for their interests. It's all self-selected, presumptuous volunteers. These are people the trans either would rather speak for them or people the trans wish would just shut up, because they make things worse. But there's no way of knowing which.
Imo, ultimately, the nation's democracy and legal system are probably going to have to weigh in. Competing social constructs should be balanced, as best we Americans can, using the will of the majority. If the majority decides, for example, that the definition of "woman" (as a social construct and for legal purposes) is "having no Y chromosome," (or whatever else), the matter is effectively settled, imo. Democracy overrides. It's all we have, our individual voices expressed through democracy.
The distinction trans remains regardless of the decision by the democracy. And rights for people who consider themselves trans would not be diminished as long as they receive equal treatment under the law in accordance with their observable behavior, not their group identification.