What can you do if someone uses your credit card without permission but you do not want to press charges against them?

If she will not press charges, she is, in effect, giving him permission to use the card, albeit a bit after the fact. The card company will not consider it a fraudulent use if the card-holder is not serious about prosecuting the alleged thief.

Sue them in small claims court in a civil lawsuit. It's totally separte from criminal.