I find that BA also adjust availability depending on point of sale more than some other airlines. I've often found prices less expensive on Expedia.co.uk than on Expedia.com - given seats are open in lower fare buckets - but not being UK-based makes these difficult to book online. I've directly called the UK call centre and booked at the lower price, but this isn't always intuitive.
OP, given that you've called Expedia a few times, I'd simply initiate the chargeback with American Express at this point. You've made best efforts and are clearly in the right here, so it should be a pretty simple process.