A card of some sort would work in practice in much the same way as a blue-badge disabled parking card, or an older-person's bus pass (or for that matter, a young person's bus pass) or as a membership card for a gym. Most of the time, the owner of the card wouldn't need it, but could produce it to show their eligibility to enter places to which entry is restricted. That doesn't "intrude into their privacy" any more than it intrudes into someone's privacy to show a card which reveals their age or their disability.
Something needs to be done to distinguish the possessor of a certificate of transition. Perhaps 1% of the male population of the country may have one of these certificates and legitimately (and peacefully and respectably) enter spaces reserved for women. But how can they be distinguished from the other 99% of the male population, who should not enter? And whose motives may at worst be criminal - or at the minimum, be "for a laugh"
The elephant in the room is that men enjoy the sight of women, and will go to a lot of trouble and expense to be where women are, especially if they are scantily dressed.
Women have centuries - millenia! - of experience of the effects of male penetration of places which should have been free from them.
Folklore is full of examples. In the Old Testament of the Bible, Susannah thought she was alone, bathing in a pool in the forest, but she was being watched from behind the bushes by a couple of "respected" elders of the community, who, when she lodged a complaint against them, accused her of flaunting her body to tempt them. In the traditional original fairytale, it wasn't just a kiss that wakened the unconscious Sleeping Beauty, it was a full-on rape. The wolf stalking Red Riding Hood on her visit to her grandmother wasn't hoping merely to eat her. In ancient Rome, the band of bachelor settlers organised a festival of games to bring people from the neighbouring Sabine towns. At a given signal they each grabbed one of the Sabine women, fought off their menfolk, and rode off with their captives.
Numerous accounts of student japes tell of infiltrating the girls' dorms, for to enjoy the outrage of the duennas. There have been similar recent forays into women-only spaces by young men claiming falsely to be trans (and getting abject apologies from the management for the way a member of staff treated their claims)
The lesson that should be taken from these is that it is not difficult, in an atmosphere where trans people are the ones who automatically receive the benefit of any doubt, and where there is no requirement for them to prove their entitlement for anyone at all to claim to be "really a woman" and therefore entitled to be treated as one without question even if they are 6 ft 2, broad-shouldered, narrow-hipped and muscular, bald and bearded, with a prominent Adam's apple, a voice deeper than Paul Robeson, and an obvious interest in female talent.