But I'm suggesting the organisation might argue that that they have not used the EA exemptions and it's not a single-sex facility - it's open to both sexes but only if you meet their other (non-PC) criteria.
That is completely irrelevant. Here's the law: it is illegal / unlawful / verboten to discriminate against people for any reason whatsoever.
That's the default position.
Here's the exception:
There are a number of circumstances where people have particular needs that other people do not have.
There are a number of reasons why people have been discriminated against in the past, before equalities laws made this unlawful, and they continue to suffer the consequences to this day.
For both of those reasons, it may be necessary, in certain situations and under certain circumstances, to make an exception to the no discrimination default.
It is impossible to do so without rules, because the reasons why exceptions may be necessary are different and the interests of some people may conflict with the interests of another.
Prior to 2010 we had a whole range of separate laws trying to do right by these different groups of people, but it was difficult to navigate them, especially when conflicts arose.
Enter the Equality Act 2010.
It defined the groups by defining what particular characteristic it is that may lead to discrimination or that may result in particular needs:
age;
disability;
gender reassignment;
marriage and civil partnership;
pregnancy and maternity;
race;
religion or belief;
sex;
sexual orientation.
Now under UK law you may not discriminate against anyone for any reason, unless you cannot meet the needs of people with a particular characteristic without discriminating against those without that characteristic.
So no, you cannot make up anything else in order to exclude people, because the default is that discrimination is unlawful and it would therefore be unlawful to just make something up. To make it lawful, you have to show that one of these nine protected characteristics is the reason why you need to discriminate by excluding people.
(This does not apply to certain private clubs or private life. There is no obligation for you to treat all guests in your house the same for instance. So if you want to let all left-handed people have an extra gift, feel free.)