Dice.co.uk could not accommodate the detail in my design, as evidenced by the (free) proof they provided. I was told that:
What you have to remember is although the artwork looks great on the screen, reduced down to fit within a 13.5mm circle, images can become merged. A simpler image would be better. Let me know how you’d like to proceed.
I went with Q-Workshop instead, and they were able to handle my specifications. The downside is that the process took FOREVER (to be fair, they were in the middle of fulfilling a very large order for Privateer Press) and the price was rather high. That price included the customization of all 6 faces, though. In fact, at the time they required you to have an original design for each face.