This thread contains a lot of background
https://www.mumsnet.com/talk/womens_rights/5091477-keep-prison-single-sex-closing
There were lots of issues, KPSS weren't invited to a meeting with Braverman IIRC (and other meetings with government, even on highly KPSS relevant issues) but other groups were, including SM, and didn't push for KPSS to be included, there was work that they did that was not acknowledged by other GC groups.
There were differences of opinion that were not handled well. KPSS in general were a lot more uncompromising - imo because having really spoken to and understood the harms to vulnerable women in prisons they saw that fancy policy wankery such as 'legal sex' and 'biological sex' or amendment of the GRA or even GC as a 'belief' wasn't going to actually protect women from real, actual harm and could be used - in some circumstances - to abuse them.
GC as a 'belief' that is protected doesn't really help if you're facing a man flashing your 10 year old daughter in the toilets, or indeed if you're facing with a rapist with an erection threatening you in a women's prison's showers.
It's great progress for some situations and some women, specifically not getting fired (or at least being able to go to ET for being fired) for sex realist beliefs but not all, and not for safeguarding. GC being a protected belief doesn't improve safeguarding - which needs to be based on sex and reality not inner feelings or 'rights' law. I suppose it allows women to speak up a little more about safeguarding failures without fearing being fired (though women still are, let's face it and the fear of a long drawn out, expensive ET battle whilst not being able to work is still a huge disincentive to speak up - hence the need for SEEN), but that's about it.
The SM statement when KPSS closed was worded a bit as if Kate was an employee IMO.
KPSS was coming from a safeguarding angle and SM are more human rights / EA2010 focused as typical of political lobby groups. For women and children at the pointy end, safeguarding is more important and SM just don't fully understand safeguarding (again IMO, but there's lots of evidence of them getting it wrong and I note they also didn't support Esses and SSA, who are also safeguarding focused, calling for an inquiry as noted upthread).
But I'd suggest you read what KPSS themselves wrote about why they closed, though I think they don't name names
https://x.com/noxyinxxprisons/status/1806278966203793535?s=46