How to assign mouse button to Mission Control's 'Application windows' feature?
I use a Logitech MX Master Mouse with my iMac and instead of using the included software, I use a product called ControllerMate.

In it, I created a workflow that 


detects the lower mouse button to show all desktops
issues a CommandQ to quit
issues a CommandW to close 




You can have it send any command/text that you want and I have found it much easier and more reliable than the Logitech software.  

In your case, I would assign a keyboard shortcut to Mission Control, then have ControllerMate intercept the mouse button press and then send the keyboard shortcut.

ControllerMate is free for up to 5 building blocks and then $25 for a full license.  (I personally got the full license so I could program my buttons and a second Logitech keyboard for "quick macros"