I can tell you what the RE prediction is, but I hesitate to say what the FE prediction(s) would be. I think the best way to explain the apparent rotation of the face of the moon is with a 3D simulation. It's a 3D phenomenon, and it's hard to imagine. A simple helper is to think about what happens when you cross the equator. Imagine the moon is at its peak and you are looking at it from the north. You are looking up facing south. Now imagine a person in the south looking at the same moon. They must be looking north. The 2 of you are facing each other. If the lit side of the moon is on your left, the lit side of the moon is on the southerner's right. The moon has appeared to rotate, but it's the same moon in the same place - you have rotated so that you're looking at it. The angle you rotate to look up at the moon varies with where you are on the Earth.
That's simple enough, and it explains the difference in orientation based on what azimuth the round earth observer has to face to look toward the moon (except for if you happen to be on the point of earth where the moon is at its zenith...or, if thinking dynamically, the latitude over which the moon is transiting, in which case it's not on any azimuth but directly overhead.
I get that. And it goes along with the critique of Appaulling's experiment proposal in which moon "rotation" or "roll" as I call it depends on the observer's rotation (azimuthal viewing). Like the bottom cartoon in this illustration:

This is the same "rotation" if a single observer tracks the moon during its transit or two observers each view the moon simultaneously from different locations. We know the moon appears to tilt/roll/rotate in that axis and why.
But that's not peculiar to a globe earth. On a flat plane earth, observers are going to see a different orientation of the moon depending on where they are on earth. The degree of "tilt" or the amount of difference between what observers see won't be the same, but then I don't have a baseline for globe earth to know what to compare a flat earth prediction to. All we know is that both globe and flat earth geometries can explain why the moon will appear oriented differently to different observers (or "roll" throughout the night).
Depending on what you think the FE is like and where/what you think the moon is, you'll get different answers for what you should see. I've only really considered the usual Gleason map with the moon circling over it at 3000 miles. If there are any FEs out there who have an idea about this, please share. How does the moon move in your model? Do you have a prediction for how it should look when viewed from different places? I could probably help come up with that prediction if you explain your favorite model.
But you won't get different answers if simply asking the question "will the moon appear to rotate?" On the other board, they "violently" protested what globe believers were saying must be true about flat earth. They do NOT say the moon will not appear to rotate. It will.
What I think I'm hearing for you is that the so-called apparent rotation will be different for flat earth, and we can't predict what it will be given the ambiguities of flat earth models. Okay. But I can't predict it for globe earth either. If I could, then I would have a baseline to compare any flat earth geometry against that would make the two distinguishable. I can't find a table or a calculator or a simulator anywhere that shows what the moon should look like if viewed along a particular azimuth from a particular globe location at a particular date and time. Flat earth doesn't have that, but neither does the globe. It ought to be calculate-able, I know. I just don't know how to do it and it doesn't exist anywhere that I can find.
I thought I had a handle on the apparent moon orientation from my location, but was surprised to see that even on the same azimuth, and the same elevation, the moon "tilted" from 3 consecutive nights of full/near-full moon.
So, what's to be gained from having people participate in a multi-location, simultaneous full moon image capture? We already know the moon will appear oriented differently to each observer. FE claims that is true of flat earth too. So? Where does that leave us?
What Appaullingly was pitching was something different. It was the effect of earth's curvature and the resulting "tilt" of observers on different parts of the globe with respect to what each considered "up." That is a feature that is non-existent on a flat surface. Everyone on a flat surface has the same "up."
But how does that difference translate to how the moon is oriented? I don't know. If the moon is rotating for everyone, we can't just observe rotation. We need to have a baseline and make a prediction as to how much the moon will "roll" based on where one is observing from. And be able to discriminate how much apparent rotation is due to azimuth and how much might be due to global curve.
I'm lost as to how to do that. I get what Appaullingly was proposing (I think), but I also completely understand the criticism of it from the flat earthers. I'm all for collecting images and comparing, but I need to know what we're looking for and how globe earth is predicting something that the flat earth model can't demonstrate.
I'm hoping Paul re-joins when his stint in the cornfield expires, or that he sees this and chimes in. Because as much as I have enjoyed the subject, I'm kind of frustrated myself.