In Goodwin, the European court of human rights found the UK was violating transsexuals right to privacy by not letting them change their birth certificate.
This is an Article 8 argument: Respect for private and family life. It's worthwhile becoming familiar with it since it will be one of the TA's canards they throw out. The argument goes that: sex can't be disclosed because it's private information under Article 8. But Article 8's presumption of privacy is a qualified right. In particular it's constrained by the following factors:
- national security
- public safety or the economic well-being of the country
- prevention of disorder or crime
- the protection of health or morals
- the protection of the rights and freedoms of others.
If you wish to keep your sex private from a service provider, it is incumbent on you to not use single sex services. If you use a single sex service, that service has already been demonstrated a proportionate necessary need (i.e. the equality Act's Requirement's for qualifying a single sex service are more stringent than Article 8) and thus it is automatically justified that your sex be disclosed.
That is: all people have the expected right to privacy, including disclosing their sex, whether they're trans or not. By using a single-sex service you voluntarily waive that right in order to access the service.
Sex Matters elaborates on this:
https://sex-matters.org/posts/updates/toilets-and-human-rights/
(Edited due to image failing to post)