As you can see, these transparent areas make up a substantial amount of the elements on the design, but they cannot be animated. I will need to drop back into Illustrator, create a solid, white rectangle and use the 'Pathfinder' tool, in conjunction with the blue mask, to 'cookie'-cut the areas from the white layer, thus recreating them as seperate objects I can animate.
It'll slow me down a little, as I must also put these new elements into an existing logical container - but, this will mean I can then animate them, which is obviously what I needed in the first place. Ugh!

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Update
I'm getting pretty efficient at this procedure! All mask elements now exist as white objects, which've been indexed into a logical group (shown in the image below.) I must now move these groups into the pre-existing logical containers before I import back into After Effects.

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