Green's functions for linearised gravitational waves? Green's functions are ubiquitous in physics, or any situation where one would like to solve some set of partial differential equations with boundary conditions.
It is therefore not so surprising that they occur also in General Relativity, at least for the linearised theory; see e.g. this paper in the arxiv from 1993.
Beyond that paper I have struggled to find references to other authors making use of such methods to investigate gravitational waves. Is there some reason Green's function methods are not useful/practical for this? Is it perhaps only useful in the linearised theory?
In addition, I am not conversant with bi-tensors, so if anyone is aware of an alternate formulation of the equations in the paper that would be helpful.
 A: Here are some more recent work using Green's functions in GR, 1401.1506 and 1910.02567. The first line of the abstract of the latter sums up the issue with Green's functions in GR.

The retarded Green function for linear field perturbations of black hole spacetimes is notoriously difficult to calculate.

There are various reason why calculations of the Green's function on a curved background are hard is due to the fact that the Green's function is singular for any two points connected by a null geodesic. This frustrates the computation of the Green's function using any form of harmonic decomposition. (Keep in mind that in a black hole spacetime, you may have some unexpected null geodesics wrapping around the black hole.)
As you have also already guessed, as second issue is that Green's function make sense only at the linear level. Green's functions are by definition the solution to the field equations sourced by a delta function distribution. However, if you would try to solve the Einstein equations beyond linear order with delta function source, you will quickly find that the answer must contain ill-defined products of distributions (or of distributions with singular functions). Consequently, one cannot use a Green's function formalism at second order (or at least we do not know how). (It should be noted that second order perturbation theory of GR with curved backgrounds, is still in its infancy.)
The appearance of bi-tensors seem fairly unavoidable in formulating Green's functions for GR since you are writing a Green's function for a tensor equation. (There may be some way around this by looking at the Weyl scalars only.) A good primer on bi-tensors is in part I of this living review article.
