It’s not a bad idea, but that doesn’t seem to be the way it goes in the novels (since we’ve had at least two High Septons die in ASOIAF) – for one thing, the fact that Septons surrender their last names means that the family name is long gone before they get to the position of Most Devout or High Septon, and since funerals are primarily about helping families process death, I don’t think you’d get that sort of custom.