I recently learned that the BMS supertranslations actually add hairs to black holes and physically change the metric. In such a scenario, I am very curious to know if the black hole can be overcharged into naked singularities with large enough supertranslations like RN black holes can be overcharged into naked singularities with large enough electric charges. Whether such situations would be generic or not is a different question, of course. Further, I would like to know how these BMS hairs affect the temperature of the black hole. In particular, do the black holes reach zero temperature precisely at the value of the supertranslation at which it would make it naked singularity? For example, the RN black hole becomes zero temperature right at the value of the electric charge beyond which the black hole would become a naked singularity and the same goes with the Kerr black holes.
BMS supertranslations are symmetries of an asymptotically flat spacetime. When we say that a physical system has symmetries we mean that there exist transformations that could be performed on our system without affecting the outcome of experiments.
Mass, charge, angular momentum (and consequently temperature) of a black hole are all quantities that are unaffected by the action of BMS group. So, no, BMS supertranslations could not make a black hole naked.
… BMS supertranslations actually add hairs to black holes and physically change the metric.
That is true, but such changes applied externally and to the whole system are simply transformations that would leave its measurable dynamics invariant. This is distinct from black hole hair getting “rearranged” as a result of absorption/scattering of a particle flying in from infinity. The latter is a physical interaction process that can also change mass, charge etc. as well as alter BMS charges, such process can have measurable consequences.
To make the answer more intuitive we can ask ourselves: “Can Lorentz transformations ionize hydrogen atom?” The arguments translate: Lorentz transformations change momentum, that is they change the quantum state of a relativistic system. If we have several particles, their momenta are essential if we consider, say, a scattering problem, but the application of Lorentz transformation to an entire system would not alter results of any experiment performed on it.