If I understand correctly, the OPs father had a flight booked for later in the year. He had not started travel. Due to the unfortunate death of a family member he had to bring forward the date of travel.

If the OP's father had needed to cancel the planned travel due to the death then BA would have allowed cancellation, with appropriate documentation.

However, in this case it was a matter of changing existing travel plans, bringing them forward. This incurred a change fee and fare difference to be paid. While I appreciate that it is a difficult time for anyone, I personally would never expect an airline to give me a free change on an existing ticket in such circumstances (especially bringing it forward when the fare will invariably be higher).

I have been in a very similar situation. Flights booked to visit my grandmother, but she died two weeks before I was due to be there. I just bought a new (very expensive full fare) ticket (the other one was using miles, with no availability so no scope of changing it). There are just some times in life when dealing with family events will incur unplanned costs. This is one of them. So I would suggest that the OP gives up on trying to get the change fee/fare difference refunded.