The uniform accelerated observer sees isotropic thermal radiation with no net energy flux.(Gerlach, Ulrich H. 1983 as an example) However, people naturally believe Hawking radiation causes blackhole evaporation, assuming non-vanishing energy flux for an outside observer. The only difference is blackholes do not have a uniform gravity field, but I cannot see why it causes anisotropic radiation.
By the way, even if there is some correction causing anisotropic radiation, can we still observe some thermal radiation moving from infinity to the blackhole, which is implied by the Unruh effect?