Oh, and missing parties because she doesn't like how she looks? That's not good (DSD had anorexia so that sets off alarm bells for me, possibly unnecessarily).
That's one where I would tell her Dad, but from the point of view of coming up with a plan to boost her self confidence.
Has she hobbies or interests that she needs support with?
If not, could you try to find her a hobby?
Volunteering is very difficult to find at that age, but if eg she's in a club they might like help with the younger kids?
DSD got a boost from volunteering during the DofE, and from her part time job when she was older, so anything that gives her a sense of achievement...
Also, has she done any therapy? DSD won't engage any more, but she did have therapy when younger about her relationship with her Mum as well as about the ED. I'm not sure it helped as much on the Mum front as time, friends and spending time with her extended maternal family, but if your DSD is disengaging from her friends it might be worth considering.
You can only listen when it comes to her Mum, you can't criticise. Friends and extended maternal family can criticise her Mum's behaviour without her feeling torn, but a therapist can point out if Mum's behaviour is objectively unacceptable/ damaging.