My friend's sister runs a group for fat women and I have discussed this very issue with her recently because I thought it was interesting - she was looking for donations of second hand clothes for an event they were running so I asked her what sizes they covered and how they define it. They call themselves 'fat' rather than 'overweight' and do body positivity workshops, fitness for bigger women, plus-size clothes sales etc. When they say 'fat identified' they just say it so that they don't have to put an arbitrary limit on what is 'fat' and what isn't. Some women consider themselves 'fat' at a size 18 and some women consider that a size 18 is positively slender in comparison to them at a size 30. By saying fat identified, it just rules out the possibility of size 30 woman saying 'I don't think this size 18 woman should be here because I don't consider her fat, even though her BMI is off the scale and she gets bullied for her size all the time'.
In theory, yes, that means a body dysmorphic size 8 woman could turn up, but so far it's never happened in five years, and if it did, they would put her in touch with people who could help her with her specific problem of body dysmorphia rather than encourage her to continue (and indeed, she'd get nothing out of the group anyway because she would be too small to take part in the fashion events and the types of issues they discuss with regards to the practicalities of being overweight, eg not being able to fit through turnstiles and having people harass them for their weight in the street, wouldn't affect her.
So, basically, if you have an issue with this you're really making a bit of a fuss over nothing and reading way too much into it. Best all calm down a bit, really.