Does Hawking radiation break asymptotic flatness? Basically what the question says -- there is reason to expect that, if allowed to continue long enough for the radiation to reach future null infinity, the fact that the radiation will fall off at $\frac{1}{r}$ will break the set of conditions required for the enveloping metric to stay asymptotically flat, which will, in turn, combined with the fact that the spacetime is inherently neither static nor stationary, will subsequently remove most of the nice features that enabled us to derive the Hawking radiation in the first place.
If this is true, why is it common for people to make claims like "The black hole will eventually completely evaporate?", and other statements relating to conditions where an appreciable proportion of the black hole's initial mass has left the horizon?
 A: 
the fact that the radiation will fall off at $\frac{1}{r}$ will break the set of conditions required for the enveloping metric to stay asymptotically flat

I'm not sure this is right. There are various definitions of asymptotic flatness. Older definitions were written in terms of coordinates, newer ones in terms of conformal transformations. The original motivation, as described in ch. 11 of Wald, was to accomplish for GR what had already been done for E&M. In E&M in SR, the coordinate-based requirements given by Wald are that the fields fall off like $1/r^2$ at $i^0$, but only like $1/r$ at $\mathscr{I}^+$. This is clearly designed to allow radiation.
The definition of asymptotic flatness in Wald is actually framed in a pretty restrictive context. He first gives a definition that's purely for a vacuum spacetime (not electrovac), and then remarks that the definition carries over automatically to a spacetime in which there is a vacuum in some open neighborhood of the boundary. Obviously it should be possible to extend this to a case in which the matter fields fall off fast enough, but it looks like he just wants to avoid making the already technical discussion even more technical. But the definition of asymptotic flatness for vacuum spacetimes definitely allows for spacetimes with gravitational radiation, since the ADM energy, which is only defined in asymptotically flat spacetimes, includes the energy of gravitational radiation at null infinity. (This could probably be checked explicitly by power-counting. For an asymptotically flat spacetime, the metric differs from Minkowski by $O(1/v)$, where $v$ is an affine parameter defined in the lightlike direction.)
As further confirmation that these spacetimes with Hawking radiation are asymptotically flat, you can find Penrose diagrams for them. For example, there's one in figure 2.41 in Penrose, Cycles of Time.
