So, black holes should emit not only photons, but also gravitons, axions (if they exist), and also more massive particles like electrons, protons, and even smaller black holes (with extremely small probability).
There exists a misunderstanding here, I think.It is not the black hole that is emitting the radiation, but the environment at the event horizon, before the radius of inevitable trapping of any matter or radiation.
Black body radiation is connected with electromagnetic radiation, and has been derived and fitted to experiments for photons. It is not a classical phenomenon, it is one of the main reasons that quantum mechanics was established for the study of the microcosm of atoms molecules and particles, and declared to be the underlying framework of all nature.
Hawking showed that the radiation leaving the region above the event horizon of a black hole is mathematically the same as black body radiation. but it should be kept in mind that nothing escapes the event horizon of a black hole, so the black hole itself is not a black body radiator, but the region above the event horizon and the photon sphere ( trapped photons in orbit) is.
This radiation does not come from dipoles and quadrupoles radiating but from elementary particle interactions. These can in principle be computed using feynman diagrams of all orders and with vacuum fluctuations and particle antiparticle loops. Any particles radiated will be from a quantum mechanical effect.
Let us assume that gravity is quantized and gravitons exist.
The feynman diagrams describing the interactions of gravitons with the rest of the zoo of particles have vertices which depend on a coupling constant that is orders of magnitude smaller than the strong , and electromagnetic
is the gravitational coupling constant
to be compared to 1/137 of the electromagnetic one.
As the couplings become squared even for first order interactions, considering the complexity of the feynman diagrams that will produce a real, leaving elementary particle, the probability of gravitons contributing appreciably to the radiation of a black hole is infinitesimally small.
Here is a heuristic drawing of Hawking radiation
Showing vacuum fluctuation pairs one of them falling into the hole , the other escaping. It is not a simple situation.
If we focus on the emission of gravitons then this seems paradoxical, because the power emitted in the form of gravitational waves by an object is related to the second time derivative of the quadrupole moment. But where does the fluctuation of the quadrupole moment of a black hole come from?
Hawking radiation is a quantum mechanical phenomenon. It is not the result of a variation in a collective electric or magnetic field, as far as photon emission goes, and equally it does not depend on the variations of masses as far as graviton emissions go. It will be the result of particle interactions modeled with feynman diagrams with virtual exchanges to the available fields at the horizon of the black hole and the emission of a real particle with some probability.