Where in the challenge did it say that I was supposed to do anything with moon calculations or say anything about FE?
The challenge was solely about RE justifying itself, to which it did not sufficiently do so.
Did you even read the challenge in the OP, or your comment which started it all?
"Start a thread on any topic on astronomy that you think that RE beats FE on and I'll be happy to rip you a new one."
It isn't about RE justifying itself; it is about comparing and contrasting between RE and FE and seeing which is better at describing reality; it is about RE beating FE, which it has done quite well; or if FE beats RE or if they are on equal footing.
The topic chosen was the moon, specifically its size and distance.
For the RE side, there are a few different ways to determine the size and distance, some of which rely upon the other measurement. But they are consistent across Earth. They do vary with time, because the distance to the moon varies with time.
Meanwhile, the best FE can come up with is appealing to ignorance of how light behaves over such vast distances, with no real justification for why light should magically start bending to produce observations consistent with a RE.
Without that, they do have some simple experiments to determine size and/or distance, but they produce vastly different results depending on which locations are chosen.
So I would say RE vastly beats FE.
Even just a simple consideration shows that.
Based upon the fact that all of Earth that can see the moon sees roughly the same section of the moon, just rotated, but not squished or stretched, shows that the moon must be very far away, in basically the same direction for everyone. Otherwise we should be seeing it significantly differently from different locations. If the moon was round that would be different sections of the moon visible depending on angle. If the moon was flat, then it would appear squashed along one axis as you are viewing it from an angle.
So that would mean that everyone sees it from basically the same angle, which means it must be quite some distance away from Earth, many times the size of Earth.
This then means in order to explain the apparent difference in direction, the reference, i.e. Earth's surface, must be different in different locations, i.e. Earth must be round.
So we have a RE, with simple physics, or a FE with completely unsubstantiated magic bendy light to produce observations which are consistent with a RE.
I would say that is RE beating FE.