OK, I think I understand more now, if your partner was seriously injured or ill (and your insurance company agreed) so you moved it in the first place, I think you might be able to sway your insurance company; and if not FCA complaint might sway matters; if he saught medical help, evidence of this would probably be useful...

...although I have to say I'm surprised BA have not been empathetic in these circumstances when the previous experience on FT has been the reverse (often claimed to be one of the things that BA does well). I would be tempted to call them again and tell them that the insurance say they can't help to move the booking and see what they say.