How about a combination of the two... A label on the outside that says "FAA registration number can be found under the battery". That way no lookie-loos can report you, but if it's genuinely lost, someone who doesn't know to look under the battery will be prompted to do so.
For me, my phone number will be prominently displayed - along with "reward if found" notice.
For me, my phone number will be prominently displayed - along with "reward if found" notice.