Well I didn't read the description and got stuck on the last level for an embarrassing amount of time...
I will now do what any math nerd would do when they cannot solve a puzzle: show proof that it is impossible.
First off we can notice a parity property of the vertices as it relates to the spaces they can occupy. Let us define a coordinate system X,Y,Z such that Y is vertical and a unit is equal to the side of the cube. We will color the cube's points in two distinct colors such that two points that share an edge do not share the same color. Then by moving the cube in all possible directions, we notice that any point that a vertex of a certain color occupies cannot be occupied by one of the same color. A little more detail on that one: after any move, a vertex either does not change position at all, or changes whether it is on the ground or not (+or-1 Y-axis position) and moves one space in the +or-1 X/Z direction. Therefore, the parity of X+Y+Z will always remain the same for said vertex, after any move. A common occurrence was for the dice to be in the position it would need to be, but translated 1 tile without rotation. This, of course, would mean the target position's vertices would be in unreachable positions. To deduce this, we need only look at one edge that is parallel to the translation. after the translation, the colors on the edges would occupy unreachable positions.
Therefore, assuming there's only one target position (4 on top, 5 to the right when standing above the gate) (proof left as an exercise to the reader) it is impossible to bring the die in a position where the stage could be beaten.
You may think this is a waste of time, but such insight can help you develop better mechanics (say, you could make an apparently useless tile that rotates your cube in place to make a similar later puzzle harder). Nice game btw :)