Your software says "Wave 2.4, ours (derekimo and me) says Wave Cordless Keyboard. It appears you have the newer wave keyboard.
As you stated, you have the Cordless Desktop® Wave Pro?
I (we) have the Cordless Desktop Wave?
Side by side comparisions of the two - Keyboard + Mice Combos Your keyboard uses the advanced 2.4GHz wireless signal which may be why you don't have the security feature as Logitech may not deem it necessary for that keyboard. That's the only thing I can think of for it not being there.
As to the initial issue, other than what I suggested, I have no clue. Sorry
Perhaps the other suggestions would be worth exploring, especially moving the reciever around.
Edit: It appears derekimo has found something of interest that you may want to look into.
Thanks to its ability to transmit data in two directions, your keyboard can save power by turning itself off when your PC is dormant. Your keyboard can even sense if there’s wireless interference and change channels to establish the most secure connection.