For a Newtonian $n$-body system, the weak Newton's 3rd law implies total momentum conservation, but not vice-versa, cf. e.g. this Phys.SE post. However for a 2-body system, which OP asks about, they are equivalent, so OP's question is essentially equivalent to:
How do we see total momentum conservation in full-fledged GR without going to the Newtonian limit?
Answer: That's a great question! Notions, such as, e.g., force, mass, momentum, energy, etc., are notoriously subtle in GR. For a generic spacetime, there is no satisfactory definition of a gravitational stress-energy-momentum tensor, only a pseudotensor.
For an asymptotically flat spacetime, one may define an ADM energy-momentum 4-vector, which plays the role of conserved total energy-momentum associated with the full spacetime (incl. probes).
This answers OP's question in principle, but may not be completely satisfactory: We can easily associate localized energy-momentum to each point-probe$^1$ in the spacetime, but it is less clear how to give an independent definition for the energy-momentum of spacetime minus the probes (other than to declare it to be the difference). I.e. translated back to OP's problem: We don't seem to have an independent definition of Earth's energy-momentum by itself, even if we for simplicity assume that Earth is a black hole with a point-like singularity/matter-distribution.
$^1$ Our notion here of a point-probe is a point particle that can be assigned a localized energy-momentum, but unlike a test particle, it can backreact on the spacetime. The notion of probes is not really essential for the discussion. To emulate OP's 2-body system without using probes, consider instead 2 black holes with point-like singularities/matter-distributions. There doesn't seem to be a well-defined notion of energy-momentum associated to each individual black hole. Their individual energy-momenta are fuzzy/delocalized/ill-defined.