Okay, I looked into it a little further and this is what I've found (anyone feel free to correct me if I'm mistaken)
You can add the color "die" (or whatever you want to call it) to your Color Books under System Overview in the Generic Press module. There you can add the color, and set its parameters so that whenever Apogee sees a color with that name, these parameters are applied (in theory.) I say "in theory" because in practice, Apogee doesn't hold onto the Transparent setting when processing a file that uses the die color. I have to edit the plan and make it transparent in the Trapping module under Trapping Orders and Densities. Even if I set the die color to Transparent in InDesign it doesn't hold unless I do the above step.
So, in summary, here's what you need to do: Set the dieline to Overprint in the native app or PitStop; edit the Job Plan's Trapping module to set the die color to Transparent and that will keep the dieline overprinting and nothing will trap to it, but everything underneath it will trap correctly.
Does anyone know why the Transparent setting on the ink doesn't hold?