In a previous job, in order not to have trouble with the US, we had to make a pinky swear not to be paying/sending money/goods/anything not just to Russia, but to places where Russia could be, erm, doing shady business.

Considering that Georgian banks have been sanctioned (VTB, before they closed) and that Georgian nationals have been sanctioned, and that Georgian banks aren't playing ball with that, BA might be a tad wary of giving cash there as it might be seen under a bad light.

I'm sure lawyers will chip in a moment, but in the meantime... do you know anyone trustworthy, based in the UK/Europe, who could receive the payment on your behalf? BA's latest comms are pretty clear: they won't pay until you do so, however frustrating this might be.