As a data point, I use Keyboard Maestro for my extra Logitech mouse buttons, since I refuse to install the terrible Logi software.
I love that Logitech has three apps. Logi+, Logi, Ghub. My keyboard only shows up in Ghub. My mice only show up in Logi/Logi+..
Which the mouse I bought is supposed to work with, at least according to some people on the internet, but does not for me.