I know it sounds awful to say, but it just wasn't a big enough story for the big media outlets to send reporters out to, if they were even aware of it happening. That's not to say that woman's rights/single sex spaces etc aren't considered newsworthy, just that the specific story was not national news and the local branch (again, if they were aware of it) likely wouldn't have had the budget.
There's also the issue that the actual facts of the story are that she wasn't banned for 'identifying a trans person in the women's changing rooms', she was banned for her behavior before, during and after it. The leisure center gave the 'we don't discuss member issues' response and 'didn't answer the local papers's questions, and given that the media who did cover it don't have a quote from the trans person, presumably they don't want to speak or haven't been found, which makes balance an issue for the Beeb. The protest was tiny. There just wasn't enough to it.
GB News is a commercial entity before it's a news outlet though. A friend of mine used to freelance on production in their news room, apparently it's insane. Their editorial policy is led by how likely something is to be watched on TV and shared on social media. This is unfortunately the exact kind of story that they know will have a good chunk of their viewership either frothing with rage or howling with laughter, and most importantly, sharing on socials. It's presented, shot and edited specifically for social engagement use.
Look at how the video itself is presented: "GET AWAY from me - Trans activists fume at chaotic protest after failing to answer simple questions" - That headline doesn't tell me anything about the actual story, or women's rights/safety. It's not about that, it's about making trans activists look stupid. A professional, researched and prepared presenter/journalist sticking a mic and a rolling camera into the faces of people (some of whom clearly didn't want to talk to her) and putting them on the spot with very deliberately phrased quick fire questions isn't investigative journalism, it's 'gotcha' journalism designed to make the subject look thick.
And it worked! 180,000 people watched that video which made them about $5,000 from YouTube plus whatever they got from the ads plastered all over it.
But yeah, that's why it's on GB News not the Beeb. Now if it was a national chain of gyms that had, for example, issued a policy banning people from complaining about men entering women's changing rooms as a result of that incident, that might have been picked up by the national press. Of if the protest at the leisure center had attracted a bigger crowd.