Solved the problem myself. Here's my solution, for anyone else who finds this question and needs to know.
There IS a fifth case screw. It's hidden about as carefully as Logitech could have done it, though I have no idea why they went to this much trouble. The fifth case screw is inside the battery compartment, UNDERNEATH the sticker that lays out the battery pattern. The screw is beneath the "5" in "1.5 V". As shown here:
http://wphost.spider-e.com/wp-content/uploads/2014/02/IMG_0416.jpg
You have to dig through that sticker to reveal and remove the screw.