The FE can't explain how the sun would illuminate any celestial object due to the illumination needing to be vastly different for different people, and not matching the requirements at all for a full phase cycle.
In order to have a full phase cycle you need to have a time when the object is between you and the sun, a time when the object is to the side of you (relative to the line connecting you to the sun) and a time when you are between the object and the sun.
But as these objects are typically considered to be 5000 km above Earth, there is no way for them to be between you and the sun or for you to be between the sun and them.
So you could never have a full or new object.
Consider a "full moon", with the moon over the equator.
People north of the equator and south of the equator on the opposite side of Earth would get to see the majority of the moon illuminated. The further south on the other side, or north on the side with the moon, the fuller it would appear.
People south of the equator would be seeing the back side of the moon and see a new moon, or very little illuminated. The further south the less would be illuminated.
People directly below would see a half of the moon illuminated.
So it is actually far worse for the FE than you think.
Also related to the alleged FE solution is why just those planets? Why don't the stars also orbit the sun? Why not Earth as well or the moon as well? The sun and moon are roughly the same size, so why don't objects also orbit the moon? What causes the planets to orbit the sun in the first place?
It doesn't make sense at all. They seem to just be taking the scientific explanation for the retrograde motion and dumping it in to FE without a second thought for how or why it would work.