There are two terms that can be used for the precious metal increased value: markup and the word used in the sentence, premium, but perhaps bounty should be chosen because you have to pay the bounty plus the metal value to receive delivery.

Private mints must also pay a processing fee from metal suppliers. It they want delivery of blanks of prescribed gauge and diameter instead of sheets or strips they must also pay a blanking fee.

Interestingly there is a coin word for just the opposite of this meaning, the antonym is agio -- a deduction from the face value of a precious metal coin because it is worn. When coins of gold or silver were valued for a transaction as in international trade -- irrespective of what country or authority issued them -- they were weighed to determine the exact value of the metal. No fiat coins were accepted then.