Several people suggested I should just pay the higher fare, and then reclaim the difference? Will that work? Given how on the phone they don't acknowledge that there is an issue I don't trust that they will when I send in my claim.

Credit card chargeback really isn't an option. The Credit Card Company will probably not do it, this is a "legitimate" transaction, the debit is of the amount that I have authorised. It may not be "right" between me and BA, but BA hasn't tricked the CC company. This isn't credit card fraud.