Applecare does not list the associated payment method

I bought applecare for my iphone at an apple store. For the payment method, I used a personal card. I am now trying to change the payment method to another card. I looked at my settings and also the apple store app on my iPhone but my original payment method is not listed (regardless, I am charged monthly).

I called customer support but they offer me to wait until the day my monthly contract ends and then call them back and switch payment methods. This is very rudimentary. I don't want to wait that long.

TOPIC IS NOT DEVELOPER FORUMS, it is AppleCare related. Of course, it is not listed as an option.