I think a women's discomfort at a trans women using a toilet cubicle next to her would probably not meet the threshold, not least because a man could also claim discomfort at the presence of a trans woman leaving trans people effectively barred from public toilets which is hardly proportionate.
Again, you are wrong. I really don't think you understand the Equality Act at all.
Schedule 3, Part 7, Paragraph 27:
27 (1) A person does not contravene section 29, so far as relating to sex discrimination, by providing a service only to persons of one sex if—
(a) any of the conditions in sub-paragraphs (2) to (7) is satisfied, and
(b) the limited provision is a proportionate means of achieving a legitimate aim.
(2) The condition is that only persons of that sex have need of the service.
(3) The condition is that—
(a) the service is also provided jointly for persons of both sexes, and
(b) the service would be insufficiently effective were it only to be provided jointly.
(4) The condition is that—
(a) a joint service for persons of both sexes would be less effective, and
(b) the extent to which the service is required by persons of each sex makes it not reasonably practicable to provide separate services.
(5) The condition is that the service is provided at a place which is, or is part of—
(a) a hospital, or
(b) another establishment for persons requiring special care, supervision or attention.
(6) The condition is that—
(a) the service is provided for, or is likely to be used by, two or more persons at the same time, and
(b) the circumstances are such that a person of one sex might reasonably object to the presence of a person of the opposite sex.
So yes, a female's discomfort at the presence of a male who identifies as trans does indeed meet the threshold and no, a male's discomfort at the presence of a male who identifies as trans does not.
Especially since under Other prohibited Conduct in Part 2, Chapter 2, Paragraph 26 Harassment, the dignity of a person is separately referenced as is the right of a person not to be subjected to unwanted conduct of a sexual nature, or to be subjected to an environment that is intimidating, hostile, degrading, humiliating or offensive.
To decide whether the presence of a male who identifies as trans creates such an environment for a female person, the court must look only at the perception of said female person and no one else's, the circumstances of the case and whether it is reasonable for her to feel this way.
As the law specifically allows for the exclusion of males from spaces we expect to be female-only, a female person's discomfort therefore more than meets the threshold.
The good news for you is that because this provision covers not only sex but also gender reassignment, this is what would allow males who identify as trans to argue for alternative facilities to be provided to them under these circumstances (harassment in male toilets).
It does not however allow them to claim the right to access female-only spaces, because this provision does not have the effect of overriding the rights of female persons to single-sex facilities. The correct and equitable solution under the existing law is the provision of alternative spaces to people who identify as trans.
As they are commonly referred to as third spaces and third spaces already exist, which are known as accessible spaces, FPFW suggested the latter be used to meet this need.
Which, if you recall, is the topic of this thread, because that suggestion did not take into account that accessible spaces were created to provide an alternative solution for people with other specific needs, that is to say needs which mean they cannot access the standard facilities provided for males and females.
As you saw, FPFW were attempting to deliver an alternative solution to people who identify as trans that took their specific needs and rights into account and so did not force them to use the facilities provided for their own sex without in the process of doing so denying everyone else the right to single-sex facilities. However, FPFW did not sufficiently consider the specific needs and rights of a third protected group, those sharing the protected characteristic of disability.
That's not - as you insinuated above - because they are against disability rights like the far right, but because they had not taken the specific needs and rights (and views) of that third protected group properly into account.
FPFW have now listened to the many people who depend on accessible facilities, have apologised for their incorrect assumptions and thoughtless suggestions and have promised to consider the specific needs and rights of protected groups other than the group of people who identify as trans and the female people sex class when making policy suggestions in the future.
You'll find the sex-based exemptions on page 162 of the PDF (or page 150 of the EqA) and the harassment section on page 26 of the PDF (page 14 of the Act) using [[https://www.legislation.gov.uk/ukpga/2010/15/pdfs/ukpga_20100015_en.pdf
this link]] to a downloadable PDF version of the official text of the Equality Act.