Well after several years of discussing this you still understand it completely wrongly.
The legal position is that anyone with the protected characteristic of gender reassignment, which means has undergone, is undergoing or is intended to undergo gender transition is protected under the Equality Act and in most cases can use spaces inline with their gender. This has been the case since 1999 and these rights were strengthened in the 2010 Equality Act. Trans people have been self-IDing into single sex spaces for decades.
This protection comes with exemptions, which apply to all strands of the Equality Act which is that discrimination is permitted if it is a proportionate means of achieving a legitimate aim, or in the case of employment if it is a reasonable requirement for the role - so a white actor could not sue if they were refused an audition to play Malcolm X for example. These are what the gender critical movement calls sex based rights, although really the rights given in the Equality Act are that you should not to be discriminated against due to sex (or gender reassignment, race, sexuality etc) except in very limited circumstances. A gender recoginition certificate does not protect you from these exemptions, neither does it grant you special rights - although it does provide protection in your legal (aquired) sex - so a trans woman with a GRC is legally a woman and could join a class action sex discrimination case against her employer over equal pay for example.
The law is a bit of a bodge in this area and the main problem is that far from trans people attempting to invade women's spaces there is virtually no case law which establishes what would be a proportionate and legitimate reason for discrimination and none which has reached a higher court. It is near certain that discriminating against trans people in toilets or enclosed changing room provision is probably illegal, and it is probably legal to discriminate against trans women in the provision of single sex services for sexual violence for example. But this should only be done on a case by case basis, as opposed to a blanket policy, and efforts should me made to avoid discrimination if possible, such as offering a service in a different way. This has been the approach taken by much of the refuge and VAWG sector, who are in most cases trans inclusive but reserve the right to invoke the exemption in some cases. And this has pretty much worked for years, but until a case comes before the courts then the legality or otherwise of discrimination relating to trans people and single sex spaces is really a bit unknown - hence the endless arguments. But gender recognition certificates are not even mentioned in the Equality Act and have little bearing on who can access single sex spaces, although they may influence other areas - such as whether you should say you are a man or a woman in the census, but again this is all untested by law.
