First of all
General relativity is time reversal invariant.
That is not strictly true. Solution of Einstein field equations locally under time reversal transform into another local solution of EFE (provided there is no matter with explicit time arrow).
Is this possible in principle assuming technical problems are solvable?
First, under time reversal, black hole becomes a white hole and its event horizon becomes past horizon (antihorizon).
Second, the largest part of technical problems is that just converging gravitational waves are not enough. One also has to provide initial conditions inside the white hole interior. White holes have past singularity and for a white hole splitting into two (white or black) holes those has to be emitted by this singularity so that they would eventually emerge in the splitting, at the exact same moment that white hole's apparent horizon/antihorizon is deformed by converging 'ring-up' gravitational waves.
Third, even if one can arrange this, the resulting solution is unstable. This instability was discovered by Eardley a few decades ago:
Another more recent (and nonpaywalled) paper:
- Barceló, C., Carballo-Rubio, R., & Garay, L. J. (2016). Black holes turn white fast, otherwise stay black: no half measures. Journal of High Energy Physics, 2016(1), 157, doi, arXiv.
The key to understanding this is the observation that even a tiny amount of matter from the outside, if it gets close enough to a white hole's apparent horizon would form a new event horizon surrounding antihorizon and preventing the splitting from occuring. Instead of white hole we now have a black hole with a slightly larger (if the perturbation was small) horizon radius. The longer the time interval between perturbation and the 'scheduled split' of white hole the smaller could be the perturbation necessary to prevent the explosion. (It is somewhat ironic that instability of explosive solution results in a non-explosion).
And if the perturbation has converted white hole into black the converging gravitational waves simply fall onto a black hole and are partly scattered partly absorbed by its growing event horizon.
Does time reversal symmetry of a black hole merger imply that a splitting of one oscillating black hole in two forward in time is impossible in the absence of infalling gravitational waves?
Not really solution without infalling gravitational waves may be possible but like any other white hole solution it would has the same kind of instability.
Actually it may be more intuitive to ask the question without time reversal:
Is it possible by providing converging gravitational waves to produce a black hole merger without any ringdown radiation?
I believe the answer is yes (just a belief on my part that noise cancelling would work generally for nonlinear GR equations). And that formulation should give us a robust solution: by providing somewhat imprecise converging waves we would receive simply a merger with suppressed ringdown modes.
The variations of this instability are present in antihorizons for other solutions of EFE: inner horizon of Kerr metric, various non-traversable wormholes (such as Einstein-Rosen bridge).