Part One of Two.
Possibly giant mirrors of very light and reflective fabric could be set up in space to reflect sunlight on the unlit hemisphere of the Moon. That would also be a very large, expensive, and extravagent project, but I haven't calculated the relative difficulty and cost.
To avoid using more mega mirrors that the necessary minimum, one sufficiently large set of mirrors in one position would be preferable. The mirrors would always have to be farther from the Sun than the moon is to reflect sunlight onto the unlit part of the Moon.
The Moon's orbit has a perigee of 362,600 kilometers and an apogee of 405,400 kilometers. So a set of mega mirrors would have to orbit at least about 410,000 kilometers farther from the Sun that Earth does to reflect light back on the moon.
When the Moon was between the Earth and the Sun, the unlit half of the Moon would mostly be the near side facing Earth, so light reflected from the mirrors beyend the Earth would hit the near side and light it up.
But when the Moon was beyond the Earth as seen from the Sun, the unlit side of the Moon would mostly be the far side facing away from Earth. So reflecting light onto it would not make the Moon look fuller as seen from Earth. Thus at least one set of secondary mirrors somewhere in space would be needed to refect the sunlight from the primary mirrors onto the near side of the Moon at those times.
If the gigantic primary mirrors were at the Earth's L2 posiiton relative to the Sun, they would orbit about 151,100,000 kilometers from the Sun, about 2,000,000 kilometers beyond the Earth's orbit, and they would take one Earth year to orbit the Sun, thus staying in position relative to the Earth. That would make the total size of the mirrors needed to keep the Moon illuminated millions of times smaller than if the mirrors were not in the Sun-Earth L2 point.
Of course L2 points are rather unstable, and the gigantic primary mirrors would have to use gigantic amounts of thrust from time to time to keep in the L2 point.
Possibly there might be mirrors in the L2 point of the Moon, orbiting at about 448,900 kilometers from the Earth. Orbiting beyond the Moon, they would reflect light onto it's far side, never seen from Earth, so they would have to be aimed past the Moon toward secondary mirrors aimed at the near side of the Moon. And of course they would be closer to the Sun than the Moon was about half the time and wouldn't be able to reflect unlight on either side of the Moon at those times.
And the use of lunar statites, reflecting sunlight to stay above a nearby planet or moon, might be considered a a method to reduce the distance between the mirrors and the target Moon and thus reduce the size of the gigantic mirrors needed.
http://arcana.wikidot.com/statite
https://www.newscientist.com/article/mg12917594-000-science-polar-satellite-could-revolutionisecommunications/
So possibly someone could design the optimum system to light the Moon with space mirrors, and calculate how the cost would compare to using solar panals and lighting on the Moon.
Part Two.
This is not an answer to the Emperor's desire.
But it is possible to design a solar system where a planet naturally always has at least one full moon visible at night. Note that it wouldn't always be the same moon(s) that were full.
But there would always be at least one full moon in the night sky, without artifical lighting, and without using enormous, fantastic, unbelievable amounts of energy to keep that moon in the right position in space.
The solar system probably would have to have been constructed by a very advanced civiliation in the past, but members of the present civilization living on the planet ages later need not know or care about that and probably won't have to pay anything to the civilization that built it millions or billions of years earlier.
See my answer to this question:
How can a moon be always full (no other phases)?