Essentially, a neurological disorder (epilepsy, narcolepsy, dyslexia all count) isn't necessarily a mental disorder.
As far as Hanako... well, that's a stronger argument, but I'm wondering if the threshold for a diagnosis would be higher in Japan than in western nations, but below that threshold, treatment still being available.
It's clear that Yamaku has mental health treatment resources - Hanako mentions using them, even, and it makes perfect sense that they would have such resources. After all, consider how many students they have that suffer from PTSD (*cough*Emi*cough*). And, more exactly applicable to Hanako's situation... they may well have experience with students that were bullied due to their disability, especially given that it's Japan we're talking about. I suspect that what was meant was that they won't accept students whose only
problem is mental health issues, or maybe even mental health issues that aren't related to their disability. And, there's probably exceptions for that everywhere - Hanako's disability isn't her burns at all, it's completely her anxiety and depression, but if you were in charge of putting her in the school most appropriate for her, where would you put her? I sure as hell wouldn't put her in a school that specialized in mental health treatment, and I went
to one (an American one, but still). She needs a place that is safe from bullying (and the bullying being due to her scars - physical damage), to heal from elementary and middle school, and regain trust in people, as part of her treatment plan. That isn't a mental health school, where she's MORE likely to be bullied for being a freak, it's a disabled school, where everyone else has physical disabilities (so she doesn't particularly stand out) and relatively stable mental health.