The part I'm finding hard to wrap my head around is that gravitational waves _are_ the disturbances in the curvature of spacetime. Wouldn't that change how they're affected by the curvature of spacetime? Maybe I'm mixing the static and dynamic aspects of the field.
A ripple of water traveling along the surface of a vortex still experiences the curvature of the vortex, even though both are disturbances. (this is a terrible analogy, please take with a ladle of salt)
Firstly, somewhat nontechnically: if you think of a binary as being two ends of a barbell with an infinitesimally thin bar and heavy weights, and put the observer in the plane of rotation and far from the binary, sometimes one of the weights on the end of the "bar" is closer, sometimes one eclipses the other so the weights line up, sometimes both are equidistant. What a pair of accelerometers at the observer reports depends on the geometry, and in particular the orbital phase. We have, in effect, a giant Cavendish experiment. If we put a large compact mass between the binary and the observer, then the light image of the binary forms a https://en.wikipedia.org/wiki/Einstein_ring . Since experimentally we know that accelerometers point to the visible (and radio and so on) image of a massive system, we would expect that gravitational radiation of all sorts gravitationally lensed just like light. We usually just make a formal analogy on that, which I'll deal with a few paragraphs further down.
Slightly technically: the "curvature of spacetime" in your question is the metric tensor field ("the metric") which fills the whole of spacetime. Yes, when we have a binary like above, the metric is dynamical. We can deliberately fix a background metric chosen for calculational or conceptual ease, and capture much of the metric dynamics as small perturbations on the background.
In this picture, we usually get gravitational waves by imposing a coordinate time on a relevant part of the spacetime, and then finding which small perturbations obey an appropriate time-dependent massless wave equation.
More physically, what this picture is saying is that in the case of a single binary, if an observer stays at one spatial location and from time to time checks its accelerometers, and they will point to the retarded positions of the objects in the binary. If you put a gravitational lense between the binary and the observer, one is free to encode the lense as a perturbation, or one can add it into the background. (The former is more popular for reasons I'll explain further below.)
Alternatively, it might help to think of gravitation as a gauge theory, wherein one can only measure potential differences between two points in spacetime rather than some absolute potential.
Let's start by defining a background wherein the potential is everywhere identical and calling that the vacuum expectation value (vev). If our background has some static spherical mass on it, we can measure a potential difference between fairly-close-to-the-mass and far-from-the-mass[1]. The potential difference is not time-dependent.
As we add any such mass we may update our vev a little, but at enormous distances from all the masses, it's only a very little, so we can largely take a value much closer to our set of masses.
But when we add in more than one mass, we lose time-independence.
If the initial conditions are picked so the masses do not orbit or twist around each other, our masses will obey Raychaudhuri's focusing theorem, and we can reason using shell theorem or Birkhoff's theorem, and expect no gravitational waves detectable outside the collapsing system. On the other hand, if we let our masses fall into orbits, we get gravitational waves.
In the focusing-only case we end up seeing vev - potential_near increasing during the collapse, where near is at a large finite radial distance (in Schwarzschild coordinates) from the collapsing mass, and we measure vev at radial infinity. In the orbiting case, vev - potential_near has a long term tendency to increase, but will vary slightly depending on the orientation of the measurement points to the binary.
A large mass -- like a galaxy cluster -- far from the source is just a region where the potential departs from vev. Helpfully, when the lense mass is that large and the GW frequency is high (e.g. near the end of a binary inspiral, or quantitatively greater than ~ 1 Hz), we can draw a direct formal analogy with electromagnetism: the geometrical optics approximation [2]. Following gravitational wave analogy with https://en.wikipedia.org/wiki/Geometrical_optics we map the difference in potential from the vev to a difference in index of refraction and let the GW plane wave refract. It's useful to remember here that GWs are modelled in a linearization of the Einstein Field Theory and so treating galaxy clusters as linear media or galaxies as nested shells of linear media is a reasonable approach. Unfortunately, in general wave optical solutions are only obtainable numerically.
Using the full GR, you have a metric with pretty much no symmetries and good luck with the calculations. Your starting point for extragalactic GW sources would be a "swiss cheese" like Einsten-Straus 1931 with multiple vacuoles and then combine that with a pp-wave, probably perturbatively, ideas which have been explored in the literature although usually in cosmological/primordial GW contexts [3]. (Compare with a more formal statement of the approach in previous paragraphs, e.g. in §1.2 at http://aether.lbl.gov/www/classes/p139/homework/hw12.pdf )
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[1] If we are sufficiently close to Newton, by splitting spacetime according to a slow-moving-compared-to-light observer not very close to any compact massive objects then we can can use : https://en.wikipedia.org/wiki/Gauss%27s_law_for_gravity#Pois...
