> what is the mechanism by which the mass of an orbiting black hole pair converts its mass into a gravitational wave?

This is an excellent question, although the formal answer is essentially "mu", or alternatively in some useful approximations of General Relativity we have a research project along the lines of "what is the mechanism that generates the (final) metric of merged black holes?".

I'll try to give you a more useful answer.

The important things are that we go from observables relating to a periodically perturbed near-Schwarzschild spacetime sourced by the inspirallers (this requires us to remove other contributors to the "true" metric, so we're left only with the contributions from the inspirallers) to a much more stable near-Schwarzschild spacetime. The energy-momentum density implied at the origin before merger is higher than that afterwards. So we can ask your question: where did that energy-momentum go?

Ultimately the search for an answer comes down to making a few choices about how to describe a collection of values that appear at various points of interest in a (nonvacuum) spacetime solution of the Einstein Field Equations of General Relativity that survive a degree of calculational simplification.

A more technical response is that by choosing how to split spacetime into space and time, and by choosing to represent spacetime curvature at every point in space (for a slice of spacetime that all has the same time coordinate) as a perturbation of a fixed background metric, one can treat changes in the metric as propagating like massless waves. This is easier when the masses sourcing the metric move slowly compared to the speed of light, and when one is dealing with the increasingly-flat metric far from the source (at least tens of wavelengths from the inspiralling objects), since one can then use linearized gravity. So we're in fairly good shape far from some black holes (or neutron stars) embedded in a galaxy a few million light-years away.

One can then decide that we are not obliged to treat the perturbations as being exclusively sourced by matter, that is, the propagating gravitational waves can induce a squash-strain on matter. With some further choices of gauge and a change to a formalism like linearized gravity, one can treat some components of the two relevant tensors (the Einstein tensor G and the stress-energy tensor T) as representing specific forms of energy, with some (e.g. kinetic energy or angular momentum) being dumped into others (e.g. gravitational potential energy).

General Relativity has at its heart a relationship between matter and spacetime curvature. The former contributes to the stress-energy tensor (T) mentioned above. The components of the tensor relate to the flux of momentum-energy between a point p and its neighbouring points in the four spacetime directions. While the tensor value itself is the same for all observers, the values of the individual components (e.g. the numerical values if you write the tensor out in 4x4 matrix form) depend on choice of coordinates, different observers have almost total freedom when it comes to choosing coordinates. Spacetime curvature is represented by the Einstein tensor G, which is a non-linear function of the metric tensor g. Omitting constant factors and indices (which range from 0 to 3 for each of the four dimension of spacetime), we can write the core of General Relativity as G = T. That is, spacetime curvature is totally determined by matter. For example, G = T = 0 is the case where there is no matter, and thus flat spacetime; that's the spacetime of Special Relativity. When we add any matter at all, we deviate from flat spacetime, however because the contribution of small amounts of matter to the stress-energy tensor is small, it can be very hard to distinguish between true flat spacetime and very very slightly curved spacetime.

While in general it is convenient to think in terms of G = T -- the Einstein Tensor and thus the metric is totally determined by the configuration of matter -- there is a history of vacuum solutions of the Einstein Field equations in which G != 0, T = 0, that is that there is spacetime curvature even without matter being present. An early example was the Schwarzschild vacuum solution ("Schwarzschild spacetime"), which is perfectly spherically symmetric about an eternal gravitational singularity. Some study revealed that for a perfectly spherical uncharged unrotating arrangement of mass gives you Schwarzschild spacetime at a bit of a distance from the mass. For slight deviations from this perfect arrangement of central matter, we still get something very similar to Schwarzschild at a sufficient distance. Indeed, if one goes to enormous distances, Schwarzschild-like spacetime also starts to look like flat spacetime -- it's "asymptotically flat".

We can then ask: at a sufficient distance from a pair of inspiralling objects, is the spacetime very similar to Schwarzschild spacetime? The answer is almost "yes". If you consider these objects as if they were a barbell with an infinitesimally thin handle connecting the two weighted ends, and tumble the barbell around the middle, to a chosen distant observer stationary with respect to the centre of mass of our inspiralling objects, sometimes sometimes one or the other weight will be spatially closer. When the observer is trying to figure out if it is in Schwarzschild spacetime or something else, and using Schwarzschild coordinates, the changing proximity of the barbell ends compared to the centre of mass will become apparent.

If we go back to my fourth paragraph, the observer can make a choice to consider the local measurements of the true metric compared to his or her wristwatch or atomic clock, and a further choice to compare that with the Schwarzschild metric. There will be a periodic change in difference from the Schwarzschild background, relating to the orbital period of the inspiralling objects sourcing the true approximately Schwarzschild metric.

Near the end of the inspiral, the "up and down" of the measurements of the metric they source follows a predictable evolution as the orbital period increases until the objects inevitably collide. The relative deviation from the perfect Schwarzschild metric also increases prior to collision. But after the collision the waves cease (the collided entities settle into a configuration that is much more like the perfect uncharged non-rotating sphere that would source a true Schwarzschild metric, a process called "balding" if the end result is a black hole (with perhaps some debris around it, depending on what the inspiralled objects were). Inevitably, measurements will show a greater Schwarzschild distance from the mass, or equivalently, that the source of the new "truer" Schwarzschild-like metric has less mass-energy at the origin of Schwarzschild coordinates than was in the pre-collision configuration.

We are pretty free to interpret these observations. Generally there is no reason to invoke any local magic rather than suggest that the reduction of energy-momentum is exactly balanced by the increase in the perturbations from Schwarzschild that all possible observers would record, and look for evidence of that.

Finally, how does one measure the deviation from a background like a Schwarzschild metric? Well, one option is to do it in the style of the https://en.wikipedia.org/wiki/Cavendish_experiment . Another is to use interferometry, LIGO and Virgo (and so on) style. Either way, the measurer is looking for a little change in the arrangement of matter "here-and-now" correlated with the big change in the arrangement in matter "there-and-then".

With some deliberate choices one can think of the change "here-and-now" as being various types of energy from "there-and-then" carried to us by gravitational radiation from the source, and calculate robustly on that basis. That picture holds up pretty well under detailed analysis given current experimental results.

Although this invites all sorts of analogies or statements like "spacetime is plastic" or "gravitational waves are real in the sense of being observer-independent", those are big stretches of the theoretical underpinnings (i.e., General Relativity). All we have is a metric that covers the whole of spacetime -- all points at all times -- that we can slice up so that we can think of the value of the metric changing over time. But generally different observers will prefer different slicings, and for a family of observers any inspiraller will shed no gravitational waves at all [consider a diagram BHA ---- x ---- BHB where x is an observer at rest at the centre-of-mass of a pair of black holes in a mutual circular orbit; x will not measure any gravitational radiation under any spacetime slicing]. So they are really a gauging effect. Conveniently, when one turns a gravitational wave into a bunch of particles, one finds that the particles almost certainly have to have the characteristics of a gauge boson.

And now I'm out of space and time. :-) Hopefully this was helpful, or at least somewhat interesting.