There's not enough room in a 4x4x4 to mod it into a Bump Cube. Of course you can just extend the puzzle on the outside, but that ruins the Bump aesthetic (no really small pieces). But what if you pillowed it instead?

Maybe someone with CAD skills could come up with a picture to show what I mean...

I think I might have missed the objective but I was able to throw together a bump 4x4 fairly easily without pillowing.

Is there still enough room in there for a mechanism? (Did you put one in the design?)

There is a mechanism in there, I just used the V-cube mech because of how easy it is and as a proof of concept (and maybe one day as an actual product ).

If you were looking to mod a 4x4 into this you would need pillowing, and if I know the edge length I could easily make another design. I would recommend a Rubik's 4x4 because there is about 5-7 mm before you hit the mech and in my design the thinnest pieces were 4 mm thick with 2 mm before hitting the center.