You can use the /menu command-line option to show the menu without having to click on the icon. See the documentation for details. Ctrl+F "right click" will find it.
Regarding the menu not appearing on right click, it's impossible to tell without details that let me reproduce the issue. Maybe something went wrong with your computer. The Clavier+ icon menu has always appeared reliably on my computer, from Windows 95 to Windows 11, and that part of the program has not changed for many years. This is the first time in 20+ years that anyone reports an issue with it.