After quite a lot of thought on how best to diffuse the edge lighting from the LEDs, I stumbled on a method that was by far the simplest, most efficient way to spread the light. You can buy lenticular sheets from ebay - they're the same thing you used to get on little animated stickers that came with cereals, or "3D" effect cards and posters. When pressed up against an image, the many hundred half-cylinder lenses that run along the sheet allow different parts of the image to be focused dependant on the viewers axis.
It turns out there's a lot of really interesting stuff that can be done with these sheets, holding the sheet (Let's call it a lens) up, each cylinder effectively collects the the light at an angle perpendicular to the alignment of the cylinders - giving an overall image that looks like a stretched out averaged version of what's behind, along one axis - With the right background, objects can be concealed in quite striking ways.

Here's a picture of an "Invisibility shield" that's concealing a person.

By pairing up two different lenticular lenses, I'm able to spread the light evenly in two axis - a third diffusing sheet may then be used to remove the small square artefacts (that almost look like pixels) that are created by two lenses. I've decided against it for a few reasons, at this point in the project, I'm trying to remind myself not to continue moving the goalposts such that a prototype never emerges, and there's a part of the slightly pixely aesthetic that I didn't entirely dislike.
WIP - Work In Progress

Arrangement of 3 of the 4 clear polished acrylic sides - strips are all 2cm deep, and rely on total internal reflection to carry most of the light from the LED strips to the inside edges
One of the light bars with LED strip underneath, 2 strips of lenticular sheet with grooves running perpendicular to one another diffuse the light

Photo taken at with very fast exposure to simulate what light decay from the light bars would look like in the infinity mirror - I suppose it makes intuitive sense that it would still be evenly lit when underexposed, but I was still very encouraged by just how well it worked, and generally feeling very smug about my solution, right up until the moment I tried to implement it...

A test of the individually addressable LEDs reacting to samples taken from the screen
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