The nasa.gov link summarizes the speculation about a large anisotropic burst of gravitational radiation from the merging smaller precursors to the runaway supermassive black hole. One is free in GR to consider this as uniformly radiated GWs carrying away residual momentum along the x^1 and x^2 spacelike axes and most of the momentum from the spacelike x^3 axis as the merging black holes "bald" their asymmetries in line with the no-hair hypothesis. (The ^1..^3 are indices in the Einstein convention).

We can generalize somewhat, to answer your more broadly worded question.

First let's start with a system losing energy-momentum via gravitational radiation (e.g. PSR1913+16, but isolated in vacuum asymptotically flat spacetime) and under time-reversal[1], so that the system is gaining energy-momentum from gravitational radiation arriving from infinity in such a way that the orbit outspirals. (This requires a highly improbable spacetime towards (time-reversed) past infinity).

In this time-reversed picture, if you alter the incoming gravitational radiation, the energy-momentum imparted to the binary system will necessarily perturb their orbit differently; instead of causing the line segment through the barycentre between the pulsars' centres-of-mass to grow along its length with carefully timed + linearly polarized GWs[2], you can cause the entire system to move in a particular direction.

Think of incoming gravitational radiation that is highly anisotropic but arriving with lucky enough timing that the GW (in a gauge in which it is a plane wave with x linear polarization seen from "above" or "below" the plane of the binary orbit[2]) is always extended along this axis by stretch-squashing the advancing and receding bodies differently. An inertial observer in this asymptotically flat spacetime at a great distance and not feeling the GWs will see the system precess.[3]

Chapter 7 (Perturbation theory) in Carroll's _Spacetime and Geometry_ in subchapters 7.4-7.6 has a reasonably good inductive view of the mathematics in the limit of weak gravity, and the time-reversal trick should be workable with the contents of the chapter.

You could introduce anisotropy in a physically plausible (barely; Hulse-Taylor's orbital period is just a few hours) way by putting the binary pulsars near a pair of carefully arranged inspiralling supermassive black holes.

Another approach would be to consider Gravitational Bremsstrahlung ("GB") (see e.g. (d) at Kovacs & Thorne [1978] at http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_quer... which has an extremely priceless admission in its final sentence.). The tl;dr of "GB" is that just as electromagnetic bremsstrahlung changes the momentum and direction of a small charge moving past a large charge with the emission of electromagnetic radiation, a small mass moving past a large mass will emit gravitional waves. One can again time-reverse in each case.

The usual case of electrons decelerating in a medium and throwing off photons under time reversal is electrons absorbing photons and accelerating in a medium. Likewise (given a suitable frame of reference and gauge, and in analogy with cyclotron radiation as a form of electromagnetic bremsstrahlung) a small mass body deflecting around a much larger mass decelerates and throws off gravitons, but under time reversal absorbs gravitons and accelerates instead.

Finally, choosing particular (families of) observers and considering things in a frame of reference constructed on their attributes, choosing different sets of coordinates and doing gauge-fixing is normal in GR; some people hate it because relating one set of observations under particular choices like these to the observations of the same system under different choices is either hard to intuit or conversely hard to solve the equations for (and sometimes both!). I hope the above isn't a total confusing mess.

[1] in case time-reversal in this context seems crazy on its face, consider this much simpler Q&A: http://van.physics.illinois.edu/QA/listing.php?id=31314&t=is...

[2] this is seriously (overly) simplified :) in the ordinary non-time-reversed picture the orbiting system is always shedding off a spectrum of gravitational waves and the plane of polarization changes twice per orbit; I instead focus on the instantaneous state where an observer above the plane of rotation only looks twice per orbit, seeing the system with one star at 12 o'clock and the other at 6 o'clock, and
[3] the whole "clock" precesses against the distant observer's cartesian coordinates with an origin constantly on herself.

P.S: there are some more article-relevant gory details at http://iopscience.iop.org/article/10.1086/421552 ("How Black Holes Get Their Kicks"). Note the authors raise the analogy with bremsstrahlung (where the masses are very different) at the bottom of page L6 - page L7.