> To study the first stars, it would stare at the same patch of sky continuously, to collect as much light from them as possible

I hate to be the one to spoil the party, but the moon rotates. The main problem with a rotating liquid mirror is that you don't get to choose the axis of the parabola, and that axis rotates as the moon orbits the Earth.

You may be able to get a decent view of stars drifting through the view, and with enough orbits you could build up a decent picture, but that's different from staring "at the same patch of sky continuously".