If the transformation is because of the light from the moon there are a few changes we could expect. First, the cycle would still be matched to the lunar cycle of about 28 days just like on earth as that cycle is driven by the relative positions of the moon, sun, and earth. If the moon is further away from the sun than earth it is lit up more, most at opposition to the earth. In contrast the brightness would go down as it enters alignment between the sun and earth, peaking with a new moon.

Second, the near lack of atmosphere would likely increase the number of days which would be sufficient to trigger the transformation. Maybe it would be 5 nights instead of 3 nights, or maybe the transformation would be more intense leading to more extreme features or even new traits.

As for another planet and moon system, earth is unique here. No other moon is as close in size to the planet it orbits with the exclusion of dwarf planets Pluto and Charon. The bulk of moons are way smaller and also orbit around planets way further away from the sun and so would be far less bright. Perhaps not enough light would be reflected for there to be any transformation, while a confluence of moons may work together to have a higher impact on occasion.

In a more science fiction setting I would imagine having reflected light cause the transformation could make things like a ring world, a spinning ring with a central reflector, interesting. The starlight reflecting off the mirrors would theoretically make the whole ring be constantly lit for wearwolves and make night, changing the mirrors to reflect no light, the only time without wearwolves.