My advice would be as well as to try and claim on your PPI you should also try and see if you've been mis sold ppi in the first place. In fact American Express may well be fobbing you off as your not covered by your PPI. If that is the case you would be entitled to a full refund which you yourself say is a 'fairly large sum' per month. It's not ususual to be mis sold, it could be said it's more usual to be sold it properly in the first place as the banks have made billions each year for the last decade.
One tactic I'd use with your bank is if they drag their heels ask them if you've been mis sold and request the data that went with your application, I think that will spark them into action.