Would Foucault's pendulum work on the moon?

There is no significance in the fact that the same face of the Moon is always oriented towards the Earth. It simply means that the rotation of the Moon around its own axis has the same period as the orbit of the Moon around the Earth.

(This wasn't always so. It can be reconstructed that the Moon previously had a faster rotation rate than it has now. Celestial bodies have a gravitational interaction (involving tidal effect) that tends to slow rotation down. When the Moon rotation was slowed down to once per month the rotation of the Moon went into tidal lock.)

Any satellite orbiting any celestial body can have any rotation rate of its own.

Some of the Earth orbiting artificial satellites have a design where the satellite extends a pole after it has inserted into orbit. The purpose of the pole is to facilitate tidal lock of the satellite.

Uneven pressure from solar wind tends to slowly change the orientation of a satellite. The tidal effect from the pole extending from the satellite is very small, but the effect from the solar wind is very, very small, so very little means are sufficient to counteract it.

There is no connection between tidal lock and relativistic physics. Tidal lock is accounted for in terms of Newtonian dynamics. There is a theoretical relativistic correction, but the difference is so small that it's totally negligible.

Foucault's pendulum would, indeed, work on the Moon - though because the rotation is much slower (a period of about 2360 ks versus about 86 ks), the pendulum will also precess that much slower. (It will also swing with a longer period, due to the reduced gravity, but that's a different matter.)

Regarding your question about whether the rotation is "real" or not in the context of general relativity, yes it is real: Foucault's pendulum is, in fact, not a free-falling body in a gravitational field. As a pendulum, it is anchored at one end, and a string connects the pendulum bob to a hook. This string exerts a real force on that bob, and if it's moving under a force, it cannot be in free fall, by definition. As the Moon rotates, the anchored tip of the string is steadily dragged ever so slightly, which pulls the bob sideways which, in turn, causes the shift in its swinging motion.

Yes, it would work. Of course, you should take into account that gravity at the Moon's surface is only 1/6 of the value on Earth and that a Moon (sidereal) day is 27 Earth days.