The simplest way I found to address it was to create a shortcut for shuffle/unshuffle function that could then be easily allocated to the remote's button. But it seems that no shortcuts for shuffle/unshuffle exist in iTunes. Therefore, below are two little scripts which, when used in conjunction with shortcuts created with iKey, allow me to activate shuffle/unshuffle using my remote.
To turn Shuffle on:
tell application "iTunes"
set shuffle of playlist "Library" to true
end tell
To turn Shuffle off:
tell application "iTunes"
set shuffle of playlist "Library" to false
end tell
In order to ensure that the scripts work correctly when listening to your whole Library, make sure you use your actual library name. This could change based on your version of iTunes, OS language, etc. (Library, My_Library, Bibliothèque (in French), etc.).
Many thanks to Vonleigh, Doug, and other anonymous users, whose contributions helped me with this problem.

