For what it is worth, I think your "transsexual and not transgender" argument is somewhat persuasive. I don't think you can reasonably argue that 1) gender is a social construct, 2) trans people have significant biological differences from the norm, and 3) trans people therefore differ as to gender. I think transsexualism is probably a more appropriate word.

I'm not convinced by the idea that, assuming arguendo gender is wholly a social construct, it can be manipulated entirely by therapy (e.g. "pray the trans away". Plenty of things are social constructs which can't easily be discarded.