...early in the story I decided I needed something really ludicrous to harp on repeatedly, and ALSO for that thing to prove to be, through a deliberately circuitous path, immensely critical to the plot. This was the bunny. I guess you could say Cage was along for the ride.
If the question is "Why the bunny?" the answer is, because there is nothing you could possibly select to be the critical linchpin of a story that is more stupid than authentic memorabilia from the film Con Air, specifically the stuffed rabbit that Cameron Poe gave to his daughter at the end. You can try to think of something dumber, but you won't succeed.