I can see the point that if dysmorphia is causing serious and prolonged psychological distress in such a case, and all other alternatives have been tried, having the surgery might be the best option. I think that applies equally to transsexuals, 'trans-abled' people, and people who want to look like dragons or cats or whatever.
'I will kill myself if I don't get surgery' should never be taken as a reason to operate, though. Either it's manipulative, or they're a genuine suicide risk and need urgent intervention for that, not surgery.
I will say though that other unnecessary cosmetic procedures for mental health reasons do happen on the NHS. An older relative of mine had a facial feature which is perfectly ordinary and unremarkable but not considered attractive by most people. She had NHS surgery after claiming it made her depressed. Family resemblance means I have the same feature - she actually told me that I should pretend that it affected my self-esteem so I could get it 'fixed' for free. (That comment affected my self-esteem more than the feature ever had, but I put up a nice feminist argument about my self-worth not being based on my looks, despite being only 17 at the time). Anyway, I was shocked that such a stretched and underfunded healthcare system catered to that sort of thing.