I heard that asking for a "bereavement" fare almost always is useless - believe it or not the airlines found people were playing the game and trying to get them even though there is no bereavement. The only way it's possible is if your doctor (or better still a hospital) can be convinced to contact the airline directly.
The other way is if you work for a large company that has an account and can therefore contact the airline at a high (-er) level than you can
However, it is always worth mentioning to the airline on booking or check-in - they can be very sensitive both on boarding and on-board if your card is marked. I know peopel who were astonished at how kindly they were dealt with through their trip at such an awful time