LIGO, Virgo, and other gravitational wave observatory collaborations forthcoming in our solar system are expected to see the gravitational wave component of a https://en.wikipedia.org/wiki/Multi-messenger_astronomy event precede that event's electromagnetic (gamma rays, light, radio waves) component. Why? Both the electromagnetic wave and the gravitational wave obey the massless wave equation, for which there is the free parameter "c". This parameter is the wave's propagation speed in vacuum. But electromagnetism couples much more strongly with interstellar and intergalactic gas and dust than gravitation does, so such intervening media slows the electromagnetic wave compared the gravitational one.

This is a handy feature, since when a high-redshift candidate event is detected by LIGO or Virgo, various telescopes can search the inferred location on the sky, looking for a trailing component. A neutron star-black hole merger, for instance, will have a such a component. So will a star falling apart in proximity to a black hole (a "tidal disruption event"). The spread for closer events isn't so big: detection of the LIGO/VIRGO G298048 (sourced about 140 million light years away, so very low redshift) event's gamma rays trailed by about about 1.7 seconds after the gravitational waves.

We can draw a direct comparison with neutrinos. Although they are not massless, and thus obey a different wave equation, they are very very very light, so we in multi-messenger astronomy we can treat them as if they effectively move at the speed of light. (In fact, supernova multi-messenger astronomy is a strong constraint on the difference between the speed of light and the speed of neutrinos).

Neutrinos also couple with gas and dust very very weakly, and so a neutrino signal and a gravitational wave signal will arrive at nearly the same time, with the electromagnetic components arriving later.

> ... curvature ... curvature of spacetime ... Gravitational fields themselves influence the propagation of gravitational fields

While you're right that different solutions of the Einstein Field Equations of General Relativity do not superpose linearly (around a Schwarzschild black hole, a very low-mass particle behaves very differently from a one with enough mass to have a gravitational self-force: https://arxiv.org/abs/0902.0573 for gory details) it's probably easy to be misled by mixing a field view of General Relativity ("GR") with a geometry ("curvature") view.

We can take an effective field theory view of GR and say that there is some chosen background (e.g. Minkowski spacetime) that is perturbed by a non-rotating point mass, the combination of the two (Minkowski + perturbation) generates the Schwarzschild spacetime. We can then add another mass, a second perturbation, and see what the combination of three (Minkowski + perturbation_1 + perturbation_2) does. This is the approach of https://en.wikipedia.org/wiki/Post-Newtonian_expansion and as can be seen in the diagram on that page, it is only valid when the two masses are fairly far apart. It is hard not to think of the perturbations as fields in the sense that you seem to be thinking about. Unfortunately this has its limits. As you bring the masses closer together (increasing compactness, moving downwards on the Y axis in the diagram), obviously wrong predictions tend to creep in, destroying one's confidence in the idea that in a system with multiple gravitating masses, each generates its own independent gravitational field which can somehow be combined (or which somehow propagate through some background).

In the most popular General Relativity reference book, Misner, Thorne & Wheeler's Gravitation, the authors discuss the expression "prior geometry", meaning some aspect of the curvature which is externally fixed or non-dynamical. General Relativity is a theory with "no prior geometry", and they make a brief argument about this. While some decades later we are much better with post-Newtonian expansion approaches (which do fix a prior geometry, which is then studied using perturbation methods), and can ignore "no prior geometry" as much more than a slogan in many cases, unfortunately we cannot do so for all of them.

For highly relativistic problems (objects moving near c; "escape velocities" near c), one must resort to the full theory of General Relativity, either solving the exact form, a good approximation (see https://pos.sissa.it/081/015/pdf), or a numerical solution where neither of the previous two forms are known or to "hide" divergences in analytical approaches.

Additionally, for speculative modelling of highly relativistic systems we may wish to require that the model enjoy the manifest https://en.wikipedia.org/wiki/Background_independence of the full theory of General Relativity, which in a practical sense means that all possible observers will agree on the point-coincidences of the system independent of the choice of observer's system of coordinates or relative motion (object "A" and object "B" are in contact at the same point in spacetime for all observers; you don't have fast-moving observers calculate them never to have been in contact; you don't have rotating observers calculating them as never-in-contact; you don't have observers in deep space disagreeing with planet-bound observers about whether "A" and "B" come into contact, etc).

Approximations instead fix some aspect(s) into a background, and in some strongly relativistic systems, one may have to introduce counter-terms ("ghosts") for families of observers that are not ideal Eulerian observers within that background.

(Einstein has a good argument about this in Chapter XXXII ("The Structure of Space According to the Theory of General Relativity") in his 1934 book, https://www.ibiblio.org/ebooks/Einstein/Einstein_Relativity.... whose "not-even-quasi-Euclidean" argument is extended in Appendix 4 and accompanied by a further fourteen pages as Appendix Chapter 5 ("Relativity and the Problem of Space") in the (not-as-freely-available) 2nd edition https://doi.org/10.4324/9780203518922 )