Spartanburg District 7 is very diverse. (Spartanburg High School) Their arts seem to be good and they have very decent test scores on their honors track.
As far as Greenville... The Governor's School is by audition only (in one of these areas: Music, Visual Arts, Theater, Creative Writing, or Dance). And it is a
boarding school. Also... you can't audition past sophomore year... so it sounds like your daughter is too old.
There are a bunch of charter schools in Greenville.
List of Charter Schools:
Greenville County Schools - Strategic Education Plan
And there's the Fine Arts Center:
Fine Arts Center, Greenville, SC