We've been through this in the US, where the left has such an open mind on trans topics their brains have fallen out. Here's the bad news.
Making trans people an issue made no dent in left wokeness. Obviously, if the right says something you just reject it wholesale. Nothing to see here.
But it worked very well on conservatives. My dear devout Catholic friend said her whole very right wing parish was very worried about men in women's toilets. They all turned out to vote, but they all do anyway, so I'm not sure how much difference it made that way.
Some of that is completely justified safeguarding concerns. But a lot of it is not remotely Gender Critical. It's Gender Straitjacket.
It's important to remember that BoJo types are not after limits on transpeople because they care about women's autonomy or rights. On the contrary. They're anything but Gender Critical. They believe in gender boxes (gender jails...) just as much as TRAs. The only difference is they want to limit who's in which one by sex. But the gender jails themselves are exactly the same.
So it would make sense never to label them GC, because that just leads to a false sense of being on the same side. They have an understanding of safeguarding, which may sometimes be useful, but what they hope to achieve is the opposite of liberation.