general relativity - How can one reconcile the temperature of a black hole with asymptotic flatness?

A stationary observer very close to the horizon of a black hole is immersed in a thermal bath of temperature that diverges as the horizon is approached. $$T^{-1} = 4\pi \sqrt{2M(r-2M)}$$ The temperature observed by a stationary observer at infinity can then be obtained through the gravitational redshift formula (see http://en.wikipedia.org/wiki/Hawking_radiation#Emission_process) to be $$T^{-1} = 8 \pi M$$ which is what is often quoted as the temperature of a black hole.

As QGR points out here in an answer to my related question here, the resulting non-zero stress-energy tensor at infinity is incompatible with the asymptotic flatness of the Schwarzschild spacetime. What exactly is going wrong here?