Fine, it's a mental health issue. But they're still a biological woman who has an issue with being described as a woman.
I accept your point that among women with PMDD, a very small number of them might feel that they are men and they may be unhappy about the word woman. Thinking about it, I was wrong - it's not that I don't believe any biological women would be unhappy about using the word woman - I believe that it is more important to use the correct terminology.
Sometimes I don't mind if people are offended or upset. I am sure that many creationists are upset by the concept of evolution, but I would rather that they were upset than that schools pretended that creationism was a valid scientific theory.
If it's a mental health issue then it's questionable whether endorsing an incorrect belief is helpful.
If somebody honestly believes that they have a gender identity that is separate to their physical body, I think they are wrong. This wouldn't matter if the acceptance of this theory - the idea that there is such a thing as a male or female identity wasn't so harmful. Insisting that women are somehow intrinsically different to men in a way that goes beyond their bodies, and failing to acknowledge and make room for people with female biology are essentially the two things that have oppressed women for so many centuries, if not millennia.
I watched the youtube video you posted, and agree that many people feel that the way they look in the mirror doesn't match how they feel inside, but if the issue is that you feel that a woman can't have short hair or wear a tuxedo or that having short hair or wearing a tuxedo makes you a man, I strongly feel that the problem is not your sex but society's narrow view of how women and men should look and behave.
I want to live in a society where we can use words clearly, and where we don't believe there is a wrong way for a man or woman to feel, behave or look.