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To start with, black holes in the GR model don't radiate. At all. Even Hawking radiation isn't predicted by GR. Rather, it is predicted by doing quantum mechanics in curved spacetime. That is, you take the GR result as background to your quantum calculations (not unlike how you calculate the atom states in standard quantum mechanics by pretending a classical Coulomb potential, ignoring the fact that the electromagnetic field is actually a quantum field; if you take into account the quantum nature of the EM field, you get slight corrections known as Lamb shift).

In other words, you start with a classical GR description of the black hole, assuming that quantum effects are negligible for its description,. Then you do a quantum field theory calculation (in particular, quantum electrodynamics) to understand what quantum fields will do in such a background spacetime. Since you find that there will be radiation carrying away energy, you conclude that thanks to energy conservation, this should mean that the black hole becomes lighter.

So you have three components for evaporating black holes:

  * GR calculations not involving quantum effects (justified by the fact that black holes are massive, and quantum effects are negligible compared to them).

  * QED calculations not involving gravitational effects beyond the curved spacetime created by the black hole (justified by the fact that gravitation is weak enough that the gravitational effects of those electromagnetic processes will have a negligible effect, so the only gravitational effect to be considered is the effect of the black hole on spacetime curvature, which is modelled classically).

  * The general principle of energy conservation (which we expect to still hold even for processes that involve both gravitation and quantum mechanics).

Now to answer the question of gravitational Hawking radiation, we need to consider that we don't yet have a confirmed theory of quantum gravitation. We don't even know for sure that gravitation really is described by a quantum theory at all, and consequently don't know yet whether gravitons exist at all.

However we can make educated guesses on the assumption that gravitation actually follows quantum laws, and that at least for small energy perturbations, quantum gravitational effects can be described with a standard quantum field theory.

If those assumptions are true, then indeed black holes should emit gravitational waves. However, given that the gravitational force is many orders of magnitude lower than the electromagnetic force, the amount of gravitational radiation should be negligible to the amount of electromagnetic radiation. By the time that gravitational radiation would become significant, the black hole should be small enough that the approximations made for calculating Hawking radiation are no longer true.

So in short, we can't know for sure if black holes radiate gravitational waves, but we can say that almost certainly those gravitational waves will be negligible for all intents and purposes, except possibly for tiny black holes where quantum gravitational effects, if they exist, will be large enough that we can't say anything about how they will behave anyway.