So just as a summary because I think some of my points haven't been organized as efficiently as they could have been. I also consulted some other friends who are knowledgeable and they brought up some points as well.
First of all, I am challenging the notion that given the information that we have from the story as shown above, that there is not enough evidence to conclusively prove that this is an 8-Dimensional structure. There is one mention of this barrier across all of the Fate/Extra universe games, and it doesn't appear anywhere else. This is most notable because a game such as Fate/Extella involves accessing the Moon Cell's power directly through the Regalia.
Secondly, in Fate/Extella we know that Velber directly threatened the whole Moon Cell. The Moon Cell tried to fight back and was unable to remove the threat, so it opted to quarantine it within the Zero Dark instead. The White Titan does not have any statements of being higher dimensional to threaten the entire thing. Most notably, it came from the Umbral Star which is a real object in the galaxy as well so it isn't from a higher plane of existence.
Thirdly, the Moon Cell being several countable infinities above 4D does not make much sense when you consider other things about the setting. For example, we know through the existence of Quantum Time Locks that it is impossible for there to be an infinite number of parallel timelines. Only a certain number of possibilities are allowed, and the rest are purged. Thus, it is impossible for there to be anything many infinities above 4D, much less 8D when we are dealing with a finite number of universes.
Finally, there is the nature of how this relates directly to Avalon. I won't contest Avalon on this thread, so we'll assume that it is accepted. Avalon is a directly stated to surpass all of the Five Magics which is a pretty big deal. This means it surpasses
Kischur Zelretch Schweinorg's Second Magic. Now to be clear, the Second Magic allows for more universe crossing than the techniques used by Archimedes in Extella. Zeltretch has the ability to not only cross between similar timelines around the same events, but to go to other worlds entirely. For example, he could use the Second Magic to go from the Fate/stay night universe to Tsukihime. Because of this, Avalon should without a doubt be stronger than the Moon Cell. The Moon Cell being countable infinities higher doesn't make much sense.
Finally, there's the fact that the Moon Cell itself doesn't exist in any realities outside of the Fate/Extra universe. A true 8 dimensional object would exist several infinities above an infinite multiverse as far as the Tiering System is concerned, and yet this isn't the case.
The Moon Cell also only scans the surface of the World. It doesn't scan the Reverse Side where the higher dimensional nature of Avalon would be located. Given that this is a device that archives all history as accurately as possible, this is more likely a limitation of the Moon Cell than anything else.
Also, I am 100% on board with BB having Higher Dimensional Manipulation and Existence. I just think this only extends so far.
