Conditions of carriage are exactly that, the conditions under which BA will carry you from A to B. Your contract is different, and that's with AMEX and doubtless you agreed to them at the point of purchase. Moreover right now it would be AMEX sitting on the money, not BA. And at the risk of repetition, the entire gamut of consumer law starts with a simple question - who did you contract with? And that can only be AMEX. I'm a bit surprised why you are not so interested in pursuing matters with AMEX since that is where consumer law protects you. And incidentally AMEX have traditionally settled MCOL cases speedily and out of court. In the quote you provided, you are a bit snookered since BA did not sell you the ticket in question, see the opening words of the CoC clause.

CEDR is usually for EC261 cases, which may apply to your initial set of cancellations, but the area that is the focus of this thread is not EC261, it's contract law and the Consumer Rights Act, which CEDR don't handle (and specifically exclude in their terms of service). However CEDR does act on cases that are about airline fairness, so you would have to develop a case based on fairness, not on contractural points.