Hi yossimills
I can understand you would be upset by this. From your point of view, all your family did was buy a ticket and expect nothing more than to get the expected service. You certainly don't expect to show up at the airport and be accused of being a credit card scammer.
The airline policies can sometimes be very harsh. I was once sitting on my normal seat on a plane when a couple ground crew came onto the plane and over to where I was sitting. In fairly loud voices they announced there was an issue with my ticket and I needed to come off the plane. As it turned out there was an issue with the travel agent's payment details, which was easily rectified and I re-boarded. I can tell you I felt mortified, but when there is a risk of non-payment the smiles vanish.
So I think your family are justified to feel upset. However I doubt the airline will give, since they will have followed policy. Your profile doesn't mention status with BA, which might have encouraged them to show goodwill. Legally I doubt they can be found liable. So you may be better off as writing the incident off as corporate inhumanity.
I hope your sister has a good outcome from her medical visit.