If I was in this situation I'd try as much as possible to sort it out over the phone. If you get an agent that is able and willing to help they may even be able to provide an email address where you can send the images. However, that is very much pot luck unfortunately. Also, in a cyncial pov, there is really no way to correlate the luggage you had with you at the time vs the luggage in your pictures.

If the phone went nowhere I'd contact my card provider to do a chargeback and then send BA an email outlining what you'd done. I have a bad feeling that even if they agreed to refund you the money it would be many weeks coming.

I know you have tried calling many times (from your original message) but I see phoning as much more likely to resolve the matter. Experience on this forum with emailed complaints would not give me confidence in any kind of useful reply.