I understand you're comfortable with "woman" to describe yourself. I understand you refuse me the courtesy of using "woman" to describe women such as myself. So much for your empathy and compassion.
"Separate but equal" separated people by race in the United States. In Gaza, it separates people by religious and cultural heritage. In Rwanda in the 1980s it was by tribal affiliation. It's a failed experiment regardless of the arbitrary way you separate people.
It is either spectacularly naive or maliciously obtuse for you to portray the plight of transgender people as an issue about bathroom use. In several countries, trans people are not afforded employment or housing - and they're forced to be beggars or whores, and live on the streets. Even in the USA job discrimination is ferocious and pervasive, forcing most trans women into a life of poverty, forcing many into survival sex work. Violence against us is constant, and police dismiss even murders against us. Up until only a few decades ago, our lot was to be drugged and institutionalized, given electric shock treatment or lobotomies. In many countries trans women are killed by their own fathers or brothers for "bringing shame to the family." Even in California I've been harassed, attacked, strangled and sexually assaulted. We can be denied basic medical care simply if the doctor has a distaste for our kind - even if we're bleeding out. When you say that we have "the same rights" you indicate a spectacular lack of understanding on this topic.
"Cis woman" is not a political term. It is a scientific term. "Transgender" means that one's gender identity does not correspond with one's assigned sex at birth. ("Trans" - across, separate, distant + gender). A term was needed to describe people who do not fall in that category, because the lazy default tended to be words such as "normal," or "real," or "regular." All of these words are othering terms, terms of marginalization - as they obviously imply that trans folk are abnormal, unreal, irregular. So scientists coined the word "cisgender" - meaning that your gender identity conforms with your assigned sex at birth ("cis" = near, adjacent, on this side of + gender). It's not political. Only TERFs consider it political and refuse to adopt it. Are you a TERF? "By their actions you shall know them..."
You say that you oppose the assertion that biological sex doesn't matter. I've never made that assertion and I'm unaware of anyone who has.
I personally cannot care less if you refuse to use a term.
Live your life and be happy.
And please let me live mine.