I'm sorry the last few years have been so awful for you. Likewise your child and all the other children who have faced difficulties, psychological or physical.
Thinking of support for these children, though, how could it possibly be helpful to support a child's belief if that belief is simply untrue or wholly incoherent?
It's something of commonplace point by now, but still: would you support a child's belief she was fat whilst actually she was starving herself and dangerously underweight? Of course not. So for a girl's belief she was a boy, mutatis mutandis.
And I'm still a bit gobsmacked at the idea of a 13-year-old telling her parent(s) something about herself in the way the OP had it, and parents taking the assertion with a particular kind of seriousness. Again, think of a child telling her mother she was fat when she was actually underweight. Similar difficulties ensue?
I don't assume the difficulties you and your child suffer are "... an unfortunate byproduct of permissive parenting". You are not to blame, that seems clear. Blame belongs squarely on those - mostly men - who for various (often nefarious) reasons have promulgated an ideology with such dreadful consequences for children in particular.
It doesn't follow from that, however, that it's in any way sensible to try to take seriously a child's incoherent belief, for instance that she is "non-binary".
This is not really about being 'gender critical' at all: that just feeds into an assessment of the truth-value of what the child in question averred. Rather it's just about plain common sense. "Mum, I'm an Oompalumpa! ... that's what I identify as!", however vulnerable, however non-neurotypical the child, calls for the same response as "Mum, I'm non-binary! ..."
No?