The issue with trans women in this scenario is this: who is a trans woman? And it's not hate to understand what we're talking about (tip: issue might not be trans people, read on).
In order for a man to be a trans woman re accessing Women’s changing rooms, all he has to do is say "I'm a woman". He does not need to wear any specific clothes, does not have to have gender dysphoria, does not have to have had surgery or be taking hormones, does not have to have shaved his beard. He can look like my dad or your brother or Dodgy Bob at the pub. His words are what is important here.
Only not even that. Nobody can challenge the idea that he's not a member of the trans community, because that's what makes someone a bigot or "TERF". And women have lost their jobs for being accused of these things, others threatened with job loss. This means if you're the shop assistant attending the changing rooms, you actually can't easily even ask if Dodgy Bob is a man as he swaggers into the changing rooms, without being "transphobic" (because you're challenging the very idea of his existence).
Then some HQs decided to simply avoid that problem by making all changing rooms unisex. So now Dodgy Bob doesn't need to even run a minimal risk of inconvenience of being questioned on his "identity", he just goes into the unisex changing rooms and does what he wants..until he's discovered spying on you.
Is Dodgy Bob trans? Well that's another toughie. As long as the trans rights movement (and it's questionable that they're actually working on behalf of trans people..) says that being trans is confirmed the moment you say you're trans (self-ID), and cannot be questioned, rather than any official process to obtain official recognition, yes, he is. We simply can't know whether he's a dodgy bloke taking advantage of the newer laxer rules, or he's a trans person who is dodgy. As long as he can speak, he can say the words "I'm a woman" just like a trans woman can. And nobody can question that.
But discussing whether Dodgy Bob is trans or not is irrelevant. The point is that women should be able to try clothes on a shop knowing that the shop isn't setting her up as an easy target to Dodgy Bobs. And the reason that it involves the word trans here at all is because the movement to allow unquestioned self-ID in social practice is what has brought about the situation where the only way to protect women from Dodgy Bobs is to wait until they've struck and been caught.
Is it good this guy was caught? Absolutely. But why did a woman have to be harmed in the first place? And as we know with Wed offenders, it's rare they're caught on their first offence. So how many other women had this guy - or Dodgy Bob - spied on before being caught?
This is it in a nutshell. There are complexities in every aspects of this. The bottom line, however, is that instead of trying to prevent Dodgy Bobs from abusing women’s bodies, the social system says Dodgy Bobs don't exist and if they do they'll be caught. Meaning prevention - safe guarding - has gone out the window.
Women do not benefit from this.
Many trans people do not benefit from this.
But Dodgy Bob is pretty fucking happy. Cos he also knows he's unlikely to get caught and if he is, it'll likely be a slap on the wrist (a whole other issue..).