1. I would not initiate a chargeback. BA could care less, it will not dispute the chargeback, but the ticket will then have an add/collect which will likely cause its cancellation. You are then stuck purchasing a new ticket at then prevailing prices.

2. You need to enlist BA's assistance, without regard to fault. Put them through the useless chargeback and nobody will help you.

3. I would not even attempt to deal with this by phone. Front line agents simply do not understand the process (and have no need) and certainly do not have the authority to do so. Issues such as this need to be researched. Thus, a short and succinct email was the best starting point. As others note, this may take a long time because the issue is non-obvious. But, if you persist and remain firm, but helpful, you may come out OK.