The other issue is that you can't ignore Earth's gravity field either. The nett result is that a "lunastationary" orbit would either be weird, or totally unstable.

The L1 and L2 points are (meta) stable solutions, but you will note that L1 is much nearer to the moon's surface than is L2. In principle, a 1-month orbit with a perilune at L1 and an apolune at L2 might be stable, but Murphy's law suggests not.

In practice, since the Moon is tidally locked to Earth, you would probably go for L1, since it requires a shorter tether and can taper faster given the higer gradient of gravity. But then you have to face up to stability of the tether and anchor in the presence of libration, which shifts the ground point by as much as 5 degrees sideways from L1 every month.

It isn't an easy problem to do on the proverbial envelope - you'll need both sides, at least!