I agree with JK Rowling. I am a woman and I'm not going to stop being a woman because it makes some people feel uncomfortable. Whatever sex someone considers themselves, I respect it wholeheartedly. However, as a black woman I have a unique experience that cannot dissapear. I don't understand why this would be offensive to trans people.
Because I study psychology, I think I understand the point you were making in part. Sex is a social construct. It' s not that it doesn't exist. Instead, you should make the point that people shouldn't be limited to their born sex. I believe that is true. We all have the right to decide who we want to be. Furthermore, people should respect one another's life choices and ensure they are treated with dignity.
However, I exist. I'm a woman and since that's not going to change, I need that to be okay too.