I am using the Vanilla Icons, in their original file-locations in my Mod-- the problem is, whilst they show up for the Vanilla Factions, they DO NOT show up at all for my Mod-added faction. Everything is identical in terms of the path, the file-location, the names, et cetera, so I really cannot see why this is happenning...

Any ideas would be appreciated, since right now, I've no clue

Does identical extend to the capital M in Menus? If my memory serves correctly, the switch between Menus, Menus80 and Menus50 requires that the names be correctly capitalized in all references, because the replacement of that part of the path is case-sensitive. If the game can't find the string to replace, it fails the path and shows nothing. However the actual folder name doesn't have to get it right, as Windows is not using case-sensitivity when looking for the file. That one character may be the only one in the entire game where case matters.

Thank-you, that fixed it

I do wonder, though, if Oblivion can't find the file because of the case, why does it not switch back to the one in the .bsa files, like it would normally? (I'm assuming it's something with how computers work, or the difference between Windows/Oblivion, but just curious)

Because for your added faction there isn't an "original". The path you provide is "invalid" so the fall-back is - no file at all.