Well, would you look at that while we were sleeping, some kindly programmer elves cobbled together some shoes for us! The
Transform.rotation page of the Language Reference has a chunk of code that "smoothly tilts a transform towards a target rotation". That sounds a lot like what we're trying to do.
Hey, I have an idea: I'll keep an eye on the door. You copy and paste this code into your game. If anyone asks, tell them "the book made me do it".