Oh, bless her. Kids can really be cruel to each other at times. 
It must be a tough call, as a parent: on the one hand, nobody wants their DD feeling they need to change themselves to fit in and feel 'accepted', especially at such a young age.
But if it's affecting her overall self-confidence, then good for you, OP, for taking her seriously and helping her find a solution.
Emphasising inner confidence and differences being totally normal is important, of course, but a good friend of mine (olive skin, very dark hair) would absolutely know how little the bigger picture matters when facing daily taunts from classmates.
She still struggles with self-esteem issues stemming back to primary school-age bullying over facial/ body hair.
Unlike you, her less-than-sympathetic Mother told her to ignore it and point blank banned her from any hair removal whatsoever until 15. Harsh.
Her 11 yr old daughter (same soft, downy moustache issue as perhaps your DD has?) uses a very gentle, non-invasive Alida pad. You apparently have to use it very minimally, and it wouldn't cope with thicker, coarser hair. Hope that, or PPs' suggestions, helps.