For what it's worth, I think you're right, although it's been a very long time since I did any mechanics. The tidal locking isn't completely perfect, so the elevator would probably have to be basically a spool of rope rather than an anchor; and the Moon's axis of rotation precesses every 18 years ish per https://en.wikipedia.org/wiki/Cassini%27s_laws, so again there would need to be quite a lot of flex in the elevator; but I don't see why it couldn't be done.