Isn't there a problem with that?
Yes, there is indeed a problem with that

The problem arises quite simple from using the wrong solution to the Einstein equations. Most of the time when people use the term "black hole" they in fact mean

**Schwarzschild** black hole, as described by the Schwarzschild metric, which is a vacuum solution to the field equations. This means we are talking about a static and stationary black hole in otherwise completely empty space, where the mass M is constant and hence the same for all observers. However, for the purpose of discussion Hawking radiation we do in fact refer to black holes which, for a stationary observer somewhere far away,

**radiate** a thermal bath of particles; and a space-time filled with radiation of any kind is most certainly

**not** empty, so we cannot use the Schwarzschild solution to describe such objects. The solution that models radiating black holes is called the

**Vaidya-Bonnet metric**, and the geometry of that space-time is a fair bit more complicated than Schwarzschild geometry, so I will not attempt to give a complete description here. The part that is relevant for us is that in the Vaidya-Bonnet metric, the total mass of the black hole becomes a function of time, and not just a constant; and since a stationary far-away observer and a freely falling observer do not share the same concept of time, they will also disagree as to the mass of the black hole at any given instant ( this is analogues to the difference in coordinate time and proper time in Schwarzschild coordinates ). In Schwarzschild space-times that would be a problem as you correctly pointed out since the mass M is supposed to be a constant, but the same is not true in Vaidya-Bonnet space-time, so the apparent paradox immediately vanishes.

Black holes are very counter-intuitive things, and most especially so once you get quantum field theories involved as well !