I agree with BlindYeo and others that 'single-sex provision' and 'single-sex services and spaces' are probably better than 'sex segregation'.
only 'single' implies segregation, as in 'singling out'
Yes, I think that's what makes it a strong term, SexMatters. It conveys the principle of segregation without using that word, which has negative connotations for many people. As well as the fact that it is in widespread use already (e.g., single-sex schools), so people instantly have a framework to slot it into.
I agree that 'same sex' isn't impactful, mainly because as you say it means 'similar or like', hence most people would only be familiar with it in the context of gay and lesbian rights, e.g., same-sex marriage.
Single-sex provision (or single-sex services and spaces) is also more precise as it emphasises that we don't want sex segregation for the sake of segregation, but in specific circumstances where it fulfils a legitimate need.
E.g.:
Title: Sex Matters! A campaign for the right to single-sex services and spaces
Intro: This campaign is about the right and freedom to choose. To have the option and provision of single-sex services, programs and spaces, where they fulfil a legitimate need. To not have that right taken away as part of universal unisex inclusion.
Subheading: Why is single-sex provision important?
It would mean the whole document would need to be rejigged, with 'the right to choose sex segregation' replaced by 'the right to single-sex provision' or 'the right to single-sex services and spaces',* and other language adjusted too, which I realise isn't very fair to you, especially when you have already produced a document that is splendidly clear. As someone noted upthread, potentially you could spend many hours rejigging this and it still wouldn't be exactly as you and everyone else wanted to it to be. That said, I've worked in marketing and I know how crucial language and framing can be when it comes to getting one's message across.
*I understand why the document was written using the language 'the right to choose' (bc we are not saying others don't have the right to unisex spaces if they want them), but the 'choice' bit is implicit in the demand for provisions, so if you change the wording then including it as well is tautological.