So I did what you said and googled Karen White, a trans woman assigned to a women's prison and admitted to raping women there and before going to prison. The problem this case demonstrates is not what you seem to want it to be. The problem is that a convicted rapist should not be in general population in any prison. She sounds like she would have been abusive and violent regardless of the population around her. By the account of more than one person in the article, it seems, too, that she was very good at talking her way out of a more restrictive prison situation. The problem seems like one of vetting convicted felons before sentencing. We know that a small population of cis women have also been convicted of rape; should they be allowed in general population? I surely don't think so.
For those who say "but cis women rapists are a small percentage of the population," well, so is Karen White. I do not see where this is part of a larger trend of trans women going to women's prison to do crimes. And, yes, the article also said she is still transitioning, so the argument that she somehow is trying to "act trans" will not fly.
But let's get back to the point, for which the Karen White case is a bit of a straw man. I refuse to see how someone's misguided decision to put her with a general population in prison has anything to do with where I can go to the bathroom. Using the White case as a reason why I or other trans women can't use the stall next to you or get dressed in the room next to you is disingenuous at best, and possibly even calculatingly dishonest.
Also, I agree with this: "I am not a sum of my monthly bleeding parts," but object to the idea that the article Rowling mischaracterized is about women specifically. The article was about menstruation and used that language because not all who menstruate are women. If we say that only women menstruate, then we are saying that trans men and trans masculine non-binary folk who menstruate are really women. That would invalidate their gender identity, which you claim here to be against. So which is it: do you think that trans men and trans masculine NB people are really just "confused" women, or do you agree that people other than *just* women can menstruate? If you take offense at the article that Rowling was talking about, then logic dictates you would think the former. If you agree that people are not the sum of their reproductive organs, that gender identity matters, then your offense at the language in the article is confusing. There is just no cogent and intellectually honest way to defend being for trans identities while failing to recognize those identities in the case of trans men and trans masculine nonbinary people.
Re: "To be honest, I am somewhat disappointed with the misogyny I see in the trans-community," please tell me how trans people wanting rights (trans men, trans women, nonbinary folk, any of us) is the least bit misogynistic. The claim that trans people venting their frustration (that you freely admit to understanding), is misogyny only works if you don't think of trans women as women. If I criticize you for being transphobic, for promoting anti-trans ideas (such as that, because one trans woman a couple years ago did something horrific, you can't trust the other several million world-wide), that, in no way, is misogynistic.
If I express my frustration against a multimillionaire with hundreds of thousands of followers for saying things like there is no discrimination against trans people, that a person's sexual characteristics--genitals, reproductive organs, etc.--are how people should be judged, and yes, that women must have wombs, vulvae, etc. to be considered women (Or the opposite, that men with wombs and vulvae cannot be men; points which you seem to disagree with), am I being misogynistic? If so, how?
By saying that your womanhood is more than just whether you menstruate, you are, in fact, disagreeing with someone, like Rowling, who says that only sexual characteristics (such as whether you menstruate) can determine if someone is a woman (that, presumably is what she means when she says "sex matters"). And by disagreeing with another woman, are you being misogynistic? Of course not. My criticism of her is not misogynistic either. I know you are probably referring to abusive language used against her (much of it by non-trans people on our behalf, and surely almost all of it out of hurt and outrage). Although verbal abuse is not something I espouse, I will not condemn someone who is hurt from lashing out at someone punching down. Nor will I call such abuse against someone who has abused an entire group of people misogyny.
I also find this last line interesting: "Because right now, watching the trans and feminism going against each other, the patriarchy are laughing their socks off and slowly undoing everything you and us fought for." I agree that division only benefits the patriarchy. However, I disagree with the characterization that this is between feminists and trans people. Most feminists are pro-trans and disagree with almost all that Rowling has said. Most feminists know that trans women are women and trans men are men, so being pro-trans is not inherently anti-sexist.
Therefore, this is nota matter of trans people and their supporters vs. feminism. This is a matter of anti-transgender people trying to co-opt feminist rhetoric while trans people and our supporters call foul. If you choose to villainize transgender folk, or ignore our very existence (make no mistake: saying that only women menstruate ignores/erases trans masuline people), then you cannot honestly say you are doing it in the name of feminism. It is transphobia, pure and simple.