Your response to Chris is very good, and yes, absolutely, gender is socially constructed. But it is easy to mock the bigots, though your did it maturely and intelligently. The bigger issue is the implications of gender as a social construction. Instead of creating another binary by attacking bigots, I invite you to discuss the greater meaning of getting out of the stultification of gender binaries and the good and bad ways of doing that. I have written two article on this:
https://medium.com/inserting-philosophy/the-problem-with-the-transgender-label-64f822db4aea
and
Can a humane discussion on this be had that gets beyond reductionism?
P.S. - 200k Twitter followers doesn't make one a "thought leader" it just signifies the appeal of simplistic sloganeering.