From NEO to GEO --- an SPS scenario

What about a tether that straddles the L1 point, with one end in the
Lunar gravity well, the other in Earth's. Weights on each end could be
raised and lowered to help stabilize it in one dimension. Would that be
sufficient or just make it worse? Unfortunately, I'm pretty sure that wouldn't work. Consider that the raising and lowering of the weights would themselves be momentum transfers, and oppositeto the ones you would want, I'm thinking. I'm pretty sure that action-reaction (Newtonian) thrusters of some kind would be needed. But my perception of this situation is that the finer your ability to detect deviations from position, and the finer your ability to thrust, the smaller your reaction mass requirements become. I know that by bouncing lasers off of Apollo reflectors on the moon, scientists can measure the moon's distance to an accuracy of a few inches. So surely keeping on top of that instability would be no major task.