The custom set of cards I had made for The (W)hole thing were a bit too thin as well. If I’m backlit, you can see through them. I anticipated this problem and had thought of a plan.
How most playing cards are made is that they have two layers and in between those two layers there’s a layer of black glue. This glue makes it so that you can’t see through the card when it’s backlit.
I’m taking two of my custom printed cards and gluing a layer of black construction paper between them. This makes it impossible for the back image to be visible when I’m back lit AND the three layers makes the cards nice and rigid.