in real life, western scientists wondered for centuries what powered the sun. Calculations showed it could burn for just a few thousand years as in biblical chronology if powered by combustion, which was not enough for the eons of the new science of geology.
If the sun was powered by slow gravitational contraction, it could shine for a few millions of years according to later calculations, but by then geologists claimed the Earth had life for hundreds of millions of years. I have read that there was once physical violence at a scientific meeting between an astronomer and a geologist because of that problem.
And finally nuclear fusion was calculated to be the way that stars shone. Now you want to find a way for moon-sized "stars" to light a planet without having the immense mass that makes natural fusion possible. Fortunately what you want is not TOTALLY impossible.
To me a vast, totally mechanized system of power generation powering lights to light the planet seems the "simplest" solution.
IMHO a fully artificial "sun satellite" would probably have to have a vast system of power generators, repair robots and automated factories to manufacture parts for generators and lights, and manufacture the repair robots, and the repair robots would have to maintain and repair everything.
And that seems to be the simplest way to go. if the Earth and moon are ejected from the solar system build a lot of fusion reactors on the Moon to power lights covering the entire surface and all aimed at the Earth to illuminate and heat the Earth.
But if you want a system that runs by itself without machinery the way a sun runs, that is a problem, since you don't want an object with the mass necessary for natural fusion.
If you only want the moon like sun satellite to shine for a measly few thousand years, the creators of it could construct it as a giant pyre in space designed to burn for thousands of years.
One way is to use fusion power to power matter transmuting devices that transmute various elements into radioactive and maybe fissionable elements. If you move enough of such elements to the moon in question they will provide enough energy per second to light and heat the planet. The radiation produced by the elements will have to be turned into electromagnetic radiation (light) and Joanna Marietti's "nuclear light bulb" suggestion shows how to do it.
Of course the more energy a radioactive isotope produces each second, the faster it usually decays to a non radioactive isotope. Thus someone considering such a system will have to calculate which elements to use and how much to have a steady light production for the necessary time since the system was created in your story.
Maybe you want the first explorers to reach the "sun satellite" to discover that the light output will soon fall below sufficient amounts and life on their planet will die, unless they can find a way to replenish the radioactive elements on the "sun satellite".
Since a rocky, Earth like planet is formed by countless collisions of large astronomical bodies, it is molten hot soon after forming. Thus your Earth like planet and any large moons would be molten hot in the beginning. But the smaller moons would cool off and form solid surfaces many millions of years before the larger planets. Thus the planets may have been able to illuminate and heat their moons, and maybe space travelers might seed the moons with life that might survive until the planets cooled.
So the moons in question would cool off and no longer naturally illuminate their planets long before their planets cool off and had solid surfaces where life might need heat and light from the moons. By the time the planets needed light and heat from molten hot moons, the molten hot moons would have cooled and no longer naturally provide heat and light.
So a highly advanced society could have artificially reheated the moon or moons of a planet so the moons would provide enough heat and light, and then seeded the planet with multi celled life forms to take advantage of that heat and light before the moons cooled off again.
It could even be possible for an extremely unlikely natural collision many millions of years after the age when they were common to reheat the moon to molten heat.
Of course reheating a moon or moons to molten heat again would be a very big task for those super advanced aliens. They would probably divert many large asteroids and comets to impact on the moon (and not on the planet, except for any comets needed to provide water and atmosphere) and reheat it. The goal would be for one impart, or many simultaneous impacts, that are almost but not quite energetic enough to shatter the moon, to provide the maximum amount of heating energy.
And the super powerful aliens might want to collide other asteroids and comets with each other to provide a new moon for the planet to replace the one that was turned into a sun.