Well great, I'm happy to go along with that. Also I don't know any trans person who wouldn't (though I'm sure there are plenty, as with everything). They are certainly aware of what genitals they are born and unhappy with, and phrases such as "assigned male/female at birth" explicitly recognise that. However, then the argument over discrimination etc should focus on gender, not sex, because it is by their gender that women (either assigned male or female at birth) are being discriminated against, not their sex. It seems to me that your argument is objecting to people taking on another sex identity (I'm not saying this is a strawman, but ime that rarely if ever happens), when (almost) all the examples we have are about gender identity.
But then I want some persuasive arguments why safe spaces, scholarships, awards, toilets, sports or whatever the issue is, should apply to sex rather than gender. I made my concession about sports. I don't see why sex (as opposed to gender) has to impinge on the other areas, though I'm open to hear your arguments, such as safe spaces, as well. (a third space as a compromise as you say, maybe - but for this I would also like some clear reasoning why this would be so necessary - as earlier posts have outlined, there's no evidence that transwomen are any danger to cis women). [Not that as a cis man it is my part to agree or disagree to any compromise here.]
But I have the feeling there is something more to all of this - women have had it bad in a patriarchal society, and the cause of this is men - that now that women have at least some amount of safe spaces and protection from discrimination through exclusive scholarships, awards, etc. And now that you have all that, you see that there are "men" - who want in on those things too. And this is unfair. I can understand this, I don't know how it feels, but I think I get it (or, correct me if I got it wrong).
When there is a strong social identity, formed through adversity, other people can't just join the club, especially when they are from the outgroup that caused the adversity in the first place. But these specific people are not responsible for the patriarchy (well I suppose maybe some of them are, because there is always an exception), and they suffer just as much from the patriarchy (though, seeing that transpeople are now everybody's outgroup, they arguably suffer even more).