I'm not sure that that system would be unstable in human timeframes since the two would be tidally locked, although it would certainly alter the engineering stresses in ways that I'm grossly unqualified to calculate. I think a portion of the cable might be under compression rather than tension? I guess it depends on the rotational speed of the whole system.
Speaking of which, substantial amounts of energy would need to be spent accelerating the spin of the Earth/space elevator system to maintain a 24-hour day/night cycle.
However, Luna's presence would perturb the whole system, either tearing it apart with tidal stresses or being ejected from the system before that could happen.
See Isaac Arthur's video about it here: https://youtu.be/dc8_AuzeYKE?t=470
For Luna or Mars, gravity is reduced and the required diameter is less. Maybe it would even be feasible to build such an elevator if it were above the Earth. But now you're building above an alien plant, so you trade one set of potentially insurmountable obstacles for another.