Here's my view, which is similar to what strawberriesaregood said.
Transwomen, obviously, aren't female at birth. But assuming that a transwoman 'passes' socially as a female post transition then they will be treated exactly the same as any other woman. Yes they've had different experiences prior to their transition, but for social purposes they are women. Transwomen who don't pass will get a lot of abuse and often (wrongly) be assumed to be gay men.
I used to have very short hair and it was amazing how often I got called 'sir' as a result (though it was usually from the back). I'm short with obviously female body fat distribution, and pretty feminine features but my hair plus jeans and trainers was enough to get me 'read' as a man. Now I don't have very short hair and it never happens. I don't get all the homophobic abuse I used to either.
I'm gender critical as well, but that means for me that I believe gender is a)entirely a social construct and as a result b)apart from the obvious biological differences, females and males are not in any way fundamentally different. We're all human being basically.
I'd like to move to an entirely gender neutral society, but that has to mean a society that isn't separated on sex as well as currently the gender you are assigned socially as depends on whether you're seen as male or female. It isn't dependent on whether you actually ARE male or female, because most of us don't walk the streets with our genitals showing, and you can't read someone's chromosomes by looking at them.
The problem is at the moment we've got a lot of institutions in society that are separated on sex. There may be some justification in terms of prisons and hospital wards - I'm not so gender critical that I don't see there can be pragmatic reasons for segregation on grounds of sex, but why not have unisex loos? It would make taking young children of the opposite sex to you to the loo a lot easier. I can think of a few cases when young boys have been raped by men in male loos. Mothers of sons and dads of daughters are currently faced with sending their child into a loo on their own after a certain age, which personally I find hugely anxiety making, unless you misuse the disabled loo.
Why does the labour party even have all women shortlists? If they want to encourage more women to become MPs there are other ways of doing it. Starting by looking at why women may not be coming forward, and how local parties are choosing candidates. If they dropped the shortlists and women's sections the issue would probably go away because there was nothing to argue about. I'm not a Tory but given they've had two female leaders and PMs, they seem to be doing better than labour without all the hoo ha. Labour's misogyny is a big problem but it goes further than trans issues - look at Corbyn's stance on Iran.
I've known a number of post transition trans women and I did see them as women, I certainly didn't view them as men. I can see problems with self definition, but they're related to single sex facilities. But I think at the moment the debate has become so heated that neither side is being particularly rational to be honest.
There IS a big issue of transitioning teenage girls it seems, and that is a concern, but again my concern there would be actual medical treatment. If a daughter of mine wanted to present as male and call herself a different name, it wouldn't bother me in the slightest. I'd be concerned about hormones and so on, but that's why we need to be clear that gender and sex are entirely separate things. At the moment we're equating presenting socially as men or women with actually being male or female. What is the problem with social transition?
This will probably all be a stretch for large swathes of society, but if you consider yourself a gender critical feminist, then in my view you do need to think about what that means. And it means being very clear about the divide between gender (social construct) and sex (biology).
I agree there are problems with the proposed changes to the GRA, but from what I've read, now Justine Greening has gone the tories are likely to quietly drop it because it was Greening's idea. They've seen how unpopular it is with the media who influence the population at large (very few people take notice of the Guardian, and they're never going to vote Tory anyway) and their position is so fragile, they can't afford to piss the media off. That's why May keeps condemning every single thing the media gets upset about.