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Should (lone) black holes emit gravitational waves?

I understand this is a chain of dubious assumptions, but I'm not sure exactly where I go wrong with this line of thought.

Thanks to Hawking, we understand that black holes radiate particles. However, due to the low temperature, the particles emitted are mostly photons (as photons are massless).

Secondly, the quantum theory of gravitons isn't perfect, but is a good enough approximation in some regions. Now, *if* that applies to the event horizon, black holes should emit gravitons.

But now this is a purely gravitational interaction, and this should be completely solvable in Einstein's equations. (and remove the quantumness) So shouldn't black holes emit gravitational waves, and lose energy in the process? as far as I understand, in the GR model, they don't.