Related but too different in details for answers to apply Why not build a space elevator starting from the moon?
As answered in the related question an elevator from surface to surface is impractical. But an elevator does not actually need to come to surface or even within atmosphere to be useful. An elevator built between an anchor point on the Lunar surface and a free hanging end point well within Earth's gravity but also well outside the atmosphere would still require much less energy and propellant to reach than going to moon. And savings in the opposite direction would be even greater.
I am fairly sure that the distance (and variable distance at that) makes this economically insane, but could it work from the engineering standpoint? What would the stresses involved be and how would they compare to available or proposed materials?
EDIT: As Thucydides indirectly implies as initially written this reduces to something that has the same answer as another question. Specifically any practical elevator starting from the moon has to extend into the gravity field of the Earth, but won't extend to the atmosphere. And that has already been answered. So as written first the answer was that it is believed practical.
This makes this question kind of pseudo-duplicate, the concept is different and there would be some resulting practical differences. But I forgot to actually ask about those differences. Fundamentally it would be a question of whether extending the elevator lower down to Earth's gravity to reduce the needed energy (and braking going down) would be practical. And seriously that would be IMHO a question of the relative economics of hypothetical launch system and elevator building. Which is probably unanswerable.