is it possible to bind a 5x5x5 and a 3x3x3 together?
3 blocks on a 3x3x3 = 4 blocks on a 5x5x5.

let me know, would be an interesting project

Preliminary sketch. Would it work?
Drawing is to scale. 5x5 cubie =x 3x3 cubie = 4/3x

If the measurements match as well as you say they do, it would not only work, but would be fairly simple to make.

Your biggest problem might be cutting around the mechanism for the extra cubie on the 5x5x5. Actually, you could just cut the 3x3x3, and glue the sucker on the 5x5. As long as the 3x3x3 is below the rotation plane of the 5th slice on the 5x5x5, it'll work fine.

While you're at it, you may as well tack on a 2x2x2 to the opposite side of the 3x3x3.

actually, i was also planning to make a 5x5x5-4x4x4-3x3x3-2x2x2 hybrid, in that order.
it would take a while, even using glue, and i only have 1 2x2 and 1 4x4.

Cool!