This sparked an idea in my head, and a couple hours (and one 5/32" hole in my finger) later, I ended up with this:
The mounting bases are stuck on 1-1/2" wooden disks I had lying around from a previous project (although I'm thinking of moving to metal washers so they can be stored using magnets). The posts are cut from 5/32" styrene tubes; the side posts are 3/4" long, while the front post is 3/8". I initially put posts on all four sides, but that made it exceedingly difficult to reach the die.

Speaking of which, the square in the center holds a 10mm die, which are pretty cheap: I ordered a lot of 30 from Amazon for $8.
Anyway, hope you all find this helpful!