Obviously, Earth's moon moves in the sky and thus would not meet your requirements. However, some moons are tidally locked with their planet, meaning that they stay in the same position in the sky. The Pluto-Charon system is an example of this phenomenon in our solar system. In this case, the moon would never rise on one side of your planet and it would never set on the other.
The system at night would look something like this, with the sun off to the left:
- At point A, the moon would be always be barely visible on the horizon. Most of the time it would be obscured by clouds, mountains, or trees.
- At point B, the moon would appear relatively high in the sky, but would never be directly overhead, It would provide notable light.
- At point C, the moon would always be directly overhead and would be very bright.
I'm guessing that you want the localization of this effect to be much smaller than a whole hemisphere, though, though I propose the following refinement of this model:
Your planet is tidally locked with a largely non-reflective moon with a single, relatively small, highly reflective surface feature directly facing the planet.
Let's say your moon has an enormous, crystalline structure in the middle of its disk:
If the crystal was shaped like a very, very shallow parabolic mirror, it would constantly focus all sunlight falling upon it directly at the planet, effectively forming a spotlight covering an area of the planet's surface equal to the area of the crystal mirror:
Point A: Same as before. Moon not normally visible.
Point B: Moon is visible in the sky as before. It's up to you on how bright it appears here, depending on how reflective the surface is.
Point C: The crystal focuses all sunlight falling onto it into a stream of parallel rays directed at the planet directly under it. It can be arbitrarily bright or localized depending on what properties you choose for the crystal (size and reflectivity).
During the day time, the sun would fall upon the opposite side of the moon, thereby not falling upon the crystal at all.
Hope you like the idea!
Note that the light would disappear during a Lunar Eclipse, but if the inclination of your satellite's orbit is high enough, this would never happen.