[Physicist]

The pendulum swings faster because it is closer to the center of gravitational attraction of the system, making gravity ever-so-slightly greater.

That doesn't work, because the Earth is in free-fall with respect to the combined sun-moon system. The only part that doesn't cancel is the gravitational gradient, AKA the tidal force, and this definitely is greater during an eclipse.

[Old Professer]

Are the pendula in question oriented in some respect to the line of earth, moon, sun?