I do honestly think that there is some confusion here between women's rights and women's safeguarding. Both are important, but they're not the same thing; and exclusive concern with women's safeguarding can be inimical to women's rights.
Sageguarding = giving some special protections to women and preserving and ensuring single-sex spaces for women.
Rights = avoiding direct and indirect discrimination against women and ensuring that most spaces are not segregated, and that both sexes can access them on an equal basis.
Of course, one can and should be concerned with both. But equating the two or considering only safeguarding runs the risk of supporting the sort of old-fashioned sexism which argues that the 'fair sex' are far too fragile and angelic to be exposed to the nasty, rough world of work, politics and study; that it's their natural vocation to care for family members, so they don't need financial or practical support; that they should dress 'modestly' to avoid tempting those nasty men who can't control their impulses; etc.
There is rather too much preoccupation, on both sides of the issue, with so-called 'TERF's. There simply aren't that many radical feminists, 'trans-exclusionary' or otherwise- and certainly not on the political Right. There are lots of anti-feminists (some of them even female), who oppose anyone who doesn't conform to strict gender roles. A few of these support the concept of transgender individuals, basically on the grounds that 'if you like football you must really be male and if you like cooking you must really be female'. But far more of them oppose trans people because they're not conforming to conventional gender roles and are blurring gender distinctions. And anti-feminists tend to congregate in right-wing political parties. Our Toriesa are actually better from this point of view than those in many countries (the top British political bigotry is xenophobia, not misogyny); but voting Tory does risk encouraging the socially conservative NatCon types and Trump supporters.