I'm confused by the United will win the battle comment. United does not have the authorization to charge any more then the $28 or whatever it was. They have two choices, first to honor the fare or second to cancel your ticket and refund the money.
Did Amex say they think United won't have any trouble putting through a charge for $500 or $600 that was is not authorized and not what the customer agreed to? If so, then you chould consider another card, but it's possible that because Amex isn't a Credit card the same rules don't apply to them, but a merchant just can't decide after the fact the price I paid was too low and charge me whatever they want to, no matter whose mistake it was in the first place.