This is probably as cheap as it is ever going to be. Certainly cheaper than cancelling and rebooking (since your ticket is not refundable).
I would jump on it now rather than wait and attempt to get a better offer.
It sounds like they are already doing you a favour by letting you change it at all.
Actually going through the fare basis is quite a task and not for the feint hearted. If you post it here, some of the boffins may be able to see what's what, but I wouldn't hold out much hope.
The exceptions BA sometimes makes for medical cases usually require you to send them evidence (letter from a doctor, death certificate etc) this isn't that sort of thing.