A couple of weeks ago, the mint added the following comment to their coin web site: The purchase of $1 coins under the Circulating $1 Coin Direct Ship Program is a cash equivalent or cash-like transaction. Ordinarily, such purchases are not eligible for credit card rewards, cash-back, cash rebate, and similar programs. Check with your card issuer for its terms and conditions.

Some people have assumed that this means the coins will be posted as a cash advance by the mint. I am not that familiar with the inner workings of coding of CC transactions but the CC companies and retailers must have some way of differentiating sales because AMEX has a statement in their T&Cs that cash equiv transactions will not earn points / miles and they were able to eliminate getting miles / points on TC purchases.

That being said, even if the mint does change to posting the coins as a cash advance, there are ways to circumvent the system. First, like others have said, think of the possibility of utilizing a gift AMEX or Visa type card. However this may not work as, at least in the case of AMEX, they say that the card can not be used for cash type transactions.

Another possible workaround is to use a bank debit card that earns miles. As long as the mint is accepting CC there should not be an issue with this approach.