This is probably one of the better threads I've read which discusses the actual outcomes of sex essentialism and what this would look like, as well as the problems of lesbophobia, and "misgendering" of butch presenting lesbian women.
Out of interest, I was in the loos with a prominent essentialist lesbian activist, who was angrily complaining about such things at an LGBT event that I went to. I could really empathise, knowing the problems that gender/sex policing can cause, so I empathised with her and it was only later that I discovered her disparaging views about trans people. And also I understand the worries that butch lesbian women have about people going along with transition rather than living as a gnc woman, which is what they are. However, that perspective isn't informed by trans men's emancipation. To me, there is room for both!
I can also really relate to having a very well tuned threat detector, having lived as a GNC male/male presenting for part of my life and having experienced varying degrees and all different colours of male violence. It really resonates with me. Transitioning bought me safety, as well as feeling comfortable in my own skin.
Because of this, I'm really really sensitive to people who might be upset by my physical appearance - although when I needed help, there just aren't any spaces where people who are tuned like me actually belong. My man-dar triggers me, and I constantly have to self-reassure that I'm actually safe. But also I'm distressed by upsetting people because of how I look (although I'm a lot more body positive thanks to finding partners who are actually ok with me).
But what I'm also sensitive to is violence and unpredictability from either sex, so I've felt unsafe with people rather than the actual sex, although it's much more often that men or male-assigned people overstep their boundaries so I understand the generalisations that people make.
As far as loosing the right to challenge goes, regardless of changes to the GRA - that right will still exist. The argument comes down more to whether someone "feels" that they are less able to challenge people who behave in an intimidating way. But even with changes to the GRA everyone still retains the right to challenge.
If people act in a predatory manner or in an intimidating manner, your legal sex will not really form any part of your defence. A plea of “your honour, I’m allowed to perv on the ladies in the ladies because I’ve got a female birth certificate” doesn't follow.
However, there still exists the distress of being forced to share spaces with people who trigger you and for that reason I'm very much against the proposed changes to the GRA. It would make my life difficult too.
The loos in my local pub have gender neutral toilets and the problem I have with them is that they aren't made from scratch - I.E, there are shared basins and cubicles and men in there freak me out. I can't brush my hair, wash my hands and male gaze/presence/risk, so I feel really threatened and get out asap. If the loos have their own sink and open onto the main space - like in cafe's, its MUCH safer.
In the same way a trans loo would petrify me unless it was built in the same way with a sink/mirror etc and opening on to a public space.
As far as other spaces go, prison wouldn't bother me personally because I'm very law abiding! But I am worried that male prisoners might self declare for nefarious intent. There needs to be somewhere else for them but also with keeping women safe. I've read that often people are put in forced solitary confinement, which is unfair too.