I can't help out on the history, or on the GR for that matter.
It is the case that the Kepler problem has not been "solved"
in GR in the closed-form sense that it has been solved in the
Newtonian system. A solution in numerical (or perturbative)
sense has been achieved, however, and an supporting example
exists in observational astronomy, the binary pulsar
PSR 1913+16, and a web search on that designation just yielded
140 hits, so there's lots of information out there.