"Gender is a social construct."
I don't necessarily agree with this. A male dog is not SOCIALLY a male dog because in dog society that is what he learned.
"The question you're posing is a false analogy."
I'm not so sure. You say "gender is a social construct". I say "gender RULES are a social construct". Completely different things.
Lastly, your argument seems to suggest (correct me if I'm wrong) that "social constructs" are either "bad" or "OK to be ignored". If that is the argument you are making, I think you are mostly wrong. I would argue that social rules are behaviors that have evolved because they have some benefit. I think it's ok to stretch them -- and over time they change -- but to abruptly ignore them all together can be damaging to a society not ready to embrace those changes.