Does the Maximum Entropy of a Black Hole Allow for Hawking Radiation?

[TriTertButoxy]

I originally posted this question in the subsection "Beyond the Standard Model." Unfortunately, nobody replied. Hopefully, someone here might be more insightful.

I read somewhere that a system consisting of a single black hole contains the maximum possible amount of entropy represented by the mass of the black hole. The third law of thermodynamics says that entropy must increase, in general, for all processes.

So, suppose I consider the Hawking Radiation -- the process by which Black Holes evaporate. If the black hole contains the maximal amount of entropy, how does Hawking Radiation occur from a thermodynamic standpoint?

 

[hellfire]

I guess that the point is that the entropy bound given by the entropy of a black hole (the Bekenstein bound) refers to a bounded region of space. The detection of Hawking radiation, on the contrary, occurs "at infinity".