Payment Method Banned for Single Chargeback

If you initiated a chargeback through American Express, that is likely due to it being a fraudulent charge. Apple will then consider that card to no longer be valid and you will need to add another card. It would make no sense for them to continue to use that payment method when the card is not considered to be a secure source of funds. AMEX recommends to always contact the merchant before disputing a charge.

Did you request a refund through Apple?

Request a refund for apps or content that you bought from Apple - Apple Support

Not sure what you mean by the App not refunding you, since they are not able to provide a refund for a purchase managed by Apple. In most cases all transactions in the App Store are final, but at times Apple may provide a compassionate refund. You will be notified if the refund was granted and if denied you will get an opportunity to appeal the decision for one final time.

That is common practice among businesses after a chargeback to restrict use of that card for future charges. If you dispute a charge with a credit card for an Amazon purchase, you would also not be able to use that card for future purchases. The only thing you can do is add a different card.