Call it what you want, Carmine or Carminic Acid, neither one acceptable to reliable kashrus agencies.
As for the Starbucks Frappuccinos (and I quote from the cRc website which has been updated) …
Any Bottles or cans bearing a “KD” or “Ches K” are accepted (Cholov Stam)
NOTE: Strawberry & Cream Frappuccino is NOT kosher!
The Very Strawberry Frappuccino isnt kosher either. The Grape Juice Conc doesn’t make it any better.