The product description for the 'Logitech MX Master 3S' does include "Middle click" with its 7 buttons.
To maybe further troubleshoot the "~LButton & RButton" hotkey problem you're having you may consider completely disabling the hotkey within QAP and instead try a AutoHotkey script to bind the "~LButton & RButton" hotkey to activate a QAP menu via command-line option. https://www.quickaccesspopup.com/can-i-d...batch-file
Script example:
QAPHotkey.ahk
Code:
#Persistent
#SingleInstance Force
#NoEnv
SetWorkingDir %A_ScriptDir%
* Make sure to change the file path if needed and make sure the exe set to run is "QAPmessenger.exe"
If you never worked with AutoHotkey before I could help more because you should be able to do a quick test with the portable version of AutoHotkey (AutoHotkey_1.1.37.02.zip) and create a shortcut to the script.