Since, from what I can tell, the stacking of the cards really just creates a single card with four attributes (two from the top card, and two from lower cards), you could probably just get rid of the whole stacking mechanic all together, and just have a large deck of cards will all of the possible combinations without much change to the way the game plays.
Another way to accomplish something similar would be dice with each of the special powers on each side. And after rolling bought dice, you have stack your dice into four columns, with the top face being your active special powers. Which would create a situation for a better rule (I think) of "use a special power, then you discard it" kind of thing, revealing the new die face on the stack.
However, I understand the need for it to be included due to the restrictions of the contest. Which is sort of part of the fun on the contest: fitting things into narrow restrictions which probably shouldn't be there in the first place.