"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."

There is only one reason to taper a space elevator. An space elevator works by using the tensional strength of material which is mostly resisting the force of gravity of the structure itself. For example, the structural strength of a large bridge is also mostly about supporting the load of the structure itself- the load of traffic by itself is a small percentage of the total load. The larger the required span of a bridge the higher the percentage of the bridge's strength will be used to support the structure rather than the traffic. Suspension bridges such as the Golden Gate use the tension strength of steel cables to support the load; whereas typical bridges use the compressive strength of material to support the load of the bridge. Personally, I identify a space elevator with using tensional strength just as I identify a suspension bridge as using the tensional of material. If you were building structure which "looked like" a space elevator but which mostly relying on using the compressive strength of material, I wouldn't call the structure a space elevator, instead I would choose a different term to describe it, say, a space tower.
I see two reasons why you would even consider building any space elevator. The first reason is that rockets are inefficent. If rockets were as efficent as a typical winch to lift loads, then I would have one less reason to build space elevators. Chemical rockets are hideously inefficent- even nuclear rockets are pretty inefficent. The second reason is that with an elevator there is a possiblity of "converting" orbital energy into some other form of energy. By which I mean you could capture the energy of some object going down to Earth and use that energy to make another object go up from Earth or do something else with energy, such as generate electricity. I see no other reason for a space elevator, other than it might look pretty.
Btw when I say efficent I mean how many, say kilowatts of energy is required to put a lb into orbit.
In an earlier post [months ago], I briefly looked at the feasiblity of a Earth space elevator in the near term (decades of time). My conclusion is that you need material not currently available [unobtainum, such as nano stuff] and/or some kind of hybrid space elevator- not relying mostly on the tensional strength of material to support the structure [such as buoyancy, compressive structures, magnetism, gravity of building material, and finally, gaps in the space elevator- and of course, all of this appears to be well above my pay grade:).
Believe or not all this nonsense is leading to the topic of a Moon space elevator; and without any more to-do, the main idea is NOT have a space elevator- something that uses tensional strength, but rather have a lunar tower (and naturally, if this was ever built this would need to called the Tower of Babel). Now, I haven't given this much thought, but your discussion of a space elevator hundreds of thousands of km long as prompted me to entertain such wild notions. And keeping in mind this frame of reference, we will start with the tower of Babel being a modest size of 600 miles tall. I pick 600 miles, because a 100 mile tall structure on Earth is *vaguely* possible, so I times that by 6 because the Moon is 1/6 the gravity. Now, the Moon has some big advantages over Earth besides it's 1/6 gravity. The obvious one being is the lack of wind or other weather. Another advantage is the Moon's gravity would weaken over a shorter distance as compared to the Earth. I don't have formula handy but I believe it's dependent of radius of the body. Earth: 6378 km radius and Moon: 1738 km. So if I'm remembering correctly if you go 1738 km (1079 miles) above the lunar surface the gravity should be 1/4 of that at the surface (1/24th of earth's). And you could probably find some mountain or Mascon (or anti-Mascon) which would be favorable- somewhere on the Moon. Now, if you simply launch a rocket from the top of the tower, there would be some slight advantage. One could also accelerate on some kind track connected top the tower.
Oh, well something to think about.