It's a ggod thing for your calculations that the Moon doesn't rotate wrt Earth, otherwise you would need to calculate the stability of a tether rotating through a noticeable tidal gradient (L1 is nearer to the moon than is L2, for example.

The teher would need to survive monthly flexing.

But, don't forget the nutation of the Moon. This will give motion of the notional tether point wrt L1 (which will also move, but only in the in-line direction).

In an ideal world of circular orbits, it's easy, but in the real world, all sorts of strange complications come in.

For example, is it best to put the tether on the far side, where lunar gravity is less at the surface (due to its asymmetry). or is it best to put it on the near side where the gradient of gravity is steeper and the load drops off quicker?

You'll need to solve a few differential equations for this, or approximate them by close differences!

Enjoy...