In the extreme mass ratio case of the two body problem you can. The assumption is the larger body is minimally perturbed by the smaller body, and that all gravitational radiation can be modeled as arising from the motion of the smaller body. In this case, the smaller body can be large large (a star, a pulsar, etc) as long as the other body is e.g. a hundred thousand times larger. It is in this context, using perturbative methods, that the 'generalized equivalence principle' is demonstrated.
However, for the similar mass two body problem there is no background you can treat as minimally perturbed. Different approximation methods are needed. For this problem (similar mass two body problem), for low mass objects, when not looking for accurate corrections to Newtonian dynamics, you could use linearized equations against Minkowski background. For greater accuracy, you would use high order post newtonian approximations. However, even that isn't good enough to accurately model the events producing the strongest GW signals. Thus, for the purposes of analyzing the hoped for signals from GW detectors, full numeric solution of the field equations is what is being used.
Given the inability use a perturbative approach for the similar mass two body problem, the first issue for some kind of statement about the objects following geodesics is how to even make clear statement of what this means. I haven't been able to figure out how to do this, nor have I been able to find references to ideas on how to do this, or any results in this area. Apparently, no one reading this thread has come across anything either.