Emmy or Emme are the only two IMO that don't look made-up. Emme seems to me to be the one that doesn't seem like a nickname. :)
This. Although, like another pp said, Emme may be pronounced as "Em".
All that being said though, I think she may be asked what her "real" name is...I was asked that a lot in school (my "real" name is Christy; it's not a nickname).