I doubt that you could easily copy the brain. Everything out there is more than the sum of its parts, be it a photon weighing far more than the sum of its quarks in physics, an organic molecule being more than just hydrogen, oxygen and carbon in chemistry or a group of people being more than just a number of individuals in human culture.
So our brain is more than the sum of its neurons firing or whatever. Acknowledging that complexity exists, that there is an unfathomable gap between an entity and the sum of its parts doesn't mean you have to get esoteric or religious, when we die we are gone. But while we are alive our brain might be a bit too complex of a thing to simply download.
I think that Hawking will live on via his articles and books after he dies. No insult intended but what interest could anybody have in any of his other thoughts? What you do matters, not your supposedly rich and unfathomable inner life.