Refusing to pay is a poor idea. At a minimum, your card will be revoked. Moreover, once the dispute is resolved (even if unfavorably), it is once again a valid debt and thus most certainly reportable. Finally, I would not presume that Amex will not pursue the debt.

But, OP is far from that. However, this does point to the value of properly supporting all chargeback claims when they are initially filed. Be certain to provide:

1. Copy of the underlying e-ticket receipt which will show the amount and any fare basis which might later be needed to determine fare rules.
2. Notice of cancellation (not in dispute here).
3. Your request for a refund, making the basis for your request clear. This should include clear references to COC, fare rules, and if applicable regulatory requirements such as EC 261/2004 and 14 CFR (US DOT). No need to be fancy, but citing the specifics is helpful later.
4. Any response from the carrier. Or, if no response, a note to that effect.
5. FInally, a simple point-by-point summary of the above.

Amex can be very good about this, but as you've seen, carriers are not above submitting either false or at least misleading documentation. Thus, making your case in the first instance is critical. Finally, do not hesitate to complain to your card issuer's banking regulator if it fails to come around.