This is shocking. The part where BA state they can't trace the call because they don't record all the calls is their failing for which you have to take the fall for.
Considering you are a loyal BA traveller I'm disappointed BA aren't honouring your booking as a good will gesture.
It will need to go higher in BA as it appears those responding so far are just following procedure and can't or aren't authorised to make that decision.

I hope you can sort it out, good luck!