You can square that circle by announcing at the beginning of the course that there is going to be some assignment like that, but I'm not telling you which, because the real world doesn't.
I do agree this is a good point; trust is not something that should be simply squandered. Nevertheless, this is still a lesson that needs to be taught and so often students make it to the end without a single teacher that did.
This is ambitious. I once had a college class where the students were very upset because I decided to change the number of in-class quizzes from 5 to 4 a few weeks into the course. (The quizzes made up 10% of the overall grade.) Students hate it when you do anything even remotely weird or unexpected with assessments. Telling them that there is going to be a mystery trick assessment will just make them anxious and grumpy.