A black hole bomb draws its power from the rotation of the black hole (through a process known as superradiance). The effect of the "bomb"* on the black hole would therefore be to spin it down, converting part of its rotational energy (and therefore mass) to energy in the "bomb". The end result is a black hole with a lower spin than the one you began with, but otherwise it will come out unscathed. In fact, it will be bigger than the one you started with, despite losing mass. (This is the second law of black hole mechanics at play.)
Whether the same can be said of the <PERSON> sphere is another matter, as it is likely to melt under the enormous energy involved.
In the comments it was argued that
Physics means it should be experimentally possible to construct such a sphere, ant it is absolutely not, by the behavior of space time around a black hole. The people moving the material to construct it would end in the black hole before starting.
This depends on where you would build de mirror. From an engineering perspective it shouldn't be much harder (or easier) than building a <PERSON> sphere around an active star of the same mass, if you build it at the same radius. 43
The physical principle behind the bomb does not depend on where you place the mirror, as long as the light is directed back towards the black hole.
In fact, it does not have to be a physical mirror at all. The same effect can be obtained by considering a light (but not massless) bosonic field (such as an axion) around a rotating black hole. The mass of the bosonic field will act in its equations of motion as a natural mirror scattering the field back to the black hole, where it will be amplified again and again. Such a superradiant instability would spin down the black hole and create a boson cloud around the black hole. Overtime, such clouds would decay under emission of gravitational waves, appearing as a monochromatic source.
Search for such sources has been proposed as a way of searching for light boson fields (which could be a dark matter candidate). The non-detection of such sources already puts bounds on the existence of such fields.
*The term "bomb" may not be the best way of describing what happens. As depending on the setup the energy build-up could be quite gradual. 43