It is lawful to exclude someone with the PC of GR from a single sex space reserved for the opposite sex.
I think it's more than lawful; I think it's mandatory. But yes, that was one of the issues - she said that because there was no PC that properly captured trans-identifying children that meant trans-identifying boys couldn't be included in the GG. That's complete rubbish.
What she should have said was that the section of the EA2010 on charities doesn't permit mix-and-match membership criterion. If membership is based on more than one PC the members must hold all of them. Female and Gender Reassignment - permitted. Female or (Male and Gender Reassignment) - not permitted.
Incidentally I don't see that a child couldn't qualify as holding the PC of Gender Reassigment: anyone intending to undergo a process in the future (etc.) is covered too, which could well include children. So if she said that GR could never apply to a child that would be another error.
Then she agreed with the presenter that you could only have associations based on PCs, and that for example a club for left handed people wasn't permitted. That's a complete inversion of the law. You absolutely can have an association whose members are distinguished by some characteristic that doesn't impinge on any of the listed PCs, like, for example, their being able to write with their left hand. Discrimination for society membership or in any other way on the basis of a characteristic which is not protected is lawful. The protected characterstics are the only protected ones. That's why they're called 'protected'.
Finally I think it was a material omission in what she said, as 'the' legal expert invited on to the programme, not to mention that the GG Trustees are obliged to follow the charitable purpose in the establishing document, which is to benefit 'girls', or else they are liable from their own pockets.