is there an outlet box mounted overcurrent protection device?
Your assumption is correct. The over current device is there to protect the entire wiring system from a fault and this does not change even if you install allow voltage device at the end of the circuit. 

You can only install a over current device in an approved designed enclosure and this is regulated by NEMA in order to meet NEC requirements. 

I will mention there are methods of installing over current protection so you can reduce the size of the wiring to a device, but I don't want to release that information to any DIY since there is a better than average chance it would be installed incorrectly or abused. I believe something like that needs to be done by a licensed professional with the correct parts that can not be purchased from a standard big box or local hardware store.

So weighing price between installing a #12 or an over current protection. I would think the #12 would win out.

Stay safe.