Call up your credit card company and explain that you made a mistake tracking payments on your account that resulted in a late charge and you would like to have it reversed.
Remind them that you have never had a late payment before (if you haven’t), and that you’ve been a good customer for many years (if you have). Say you promise to take better care not to let it happen again and that you really like their card and find the use of it very consumer-friendly. Mention that you have always had exemplary service from their company (you have, haven’t you?).
If the first person you speak to on the customer-service line doesn’t respond the way you want, and will not allow the fee to be reversed, ask to speak to their supervisor. Be firm and adamant about this. If they balk, get their name and ask them the name of the manager of their department. Once you are connected to the supervisor, go through the same explanation. If they balk, ask for their supervisor and don’t let them put you off. Keep asking for the next higher-up and let that person know that you would not be adverse to closing your account and going to their competition for your credit card services. That should do the trick. The goal would be to go as high as you have to up the supervisor chain until you find someone that sees things your way. It works.