Now I've stopped laughing over "todger dodger" ... 
From the article:
part of a wider, fractious public debate around trans rights that tends to stray far beyond the legal technicalities of provisions for trans people and same-sex spaces, and into a heated discussion about the legitimacy of trans identities and the philosophical fundamentals of sex and gender.
Well, if you decide to support legislative reform that allow people to self identify into a category with certain legal protections, you are going to get questions about the parameters of those categories and what defines those categories on a philosophical level.
But equally I don’t think this would be happening with any other form of discrimination.
Of course it would. If Labour supported self-id on age, disability or maternity, the exact same thing would have occurred. People would have said: "Err, wait a minute, if you allow people to self-certify their MAT B1 form, and no one is allowed to question whether they are indeed pregnant or not, aren't you going to get people taking advantage of that for dodgy reasons?"
they are in the same place as the British public, who generally take a compassionate, “live and let live” approach to the question.
This is a huge misunderstanding of how people actually approach issues. "Live and let live" usually means "Live and let live ... so long as it doesn't impact radically upon my own life." Once stuff does, that's when people start to get annoyed, and redefining the political, cultural, and social understanding of the word "man" and "woman" certainly does impact on a lot of peoples' lives.
it’s a fundamental thing about whether we speak a language voters understand
If you are not speaking a language that voters understand, you are not a national political party; you are, at best, a think tank or, at worst, a political sect.