Well, I have to say that the first order guess at cable length was a bit unsatisfying. It shows promise for the future but no certainty yet.

I integrated the total force along a uniform density cable starting at the moons surface
and ending at a variable length. I then set this cubic equation equal to zero and solved
it.

For each of the two cases on the moon, four solutions were found, all were real and one was physically realistic. One of those was a length of zero. Integrate a finite function from A to A and get ZERO, no suprise there. If anyone want's the Mathematica notebook, just ask.

The final value was the relevant one.

For a cable from the surface of the moon towards Earth the stable length is 278,400 km.
Away from the Earth, the length is 6,456,300 km. Pretty damn long although Brad Edwards
design length for an Earth elevator cable is 110,00km. Plausably ballpark.

Just to check myself, I did a third calculation for the stable length of an Earth cable without luna. It was out of the ballpark, 938,000 km! This illustrates one of the assumptions I made, uniform linear density. Dr. Edward's plan involves significant tapering of the cable so it's thinner close to earth and thicker near the long end. Both of these contribute to the relative force on the cable away from the earth, shortening the necessary length.

The two reasons for tapering of an Earth cable are atmospheric winds and space junk. Neither are yet a problem near the moon. Perhaps I'll try a taper design that is suited to the more fundamental tensional forces the cable will feel at different lengths.

A kind of variable here is the mass of the counterweight or as I like to call it, "End of the Line Station". Having a counterweight that is a fraction of the total cable mass also shortens the necessary length. A mass at the Lagrange points (L1 or L2) makes no difference to the length. It would act like an inertial choke for any displacement wavepacket (wave) transmited by the cable.

Nick mentioned nutation as something to design for in his message below. If someone knows the amplitude and periodicity of the lunar nutation and or where to find it, I'd be appreciative.

At this point, I'm thinking the nearside cable would be the best idea as the attraction of the Earth on the end of the cable results in a shorter and more stable arangement. Also for practical reasons like DTE communication and LEO transits. Let go of the cable at EOTL Station and you're heading to Earth for a quick burn at perigee or tether capture, eh SM?