Interesting that the standalone install behaves differently. If changing the mouse gesture settings affects it, that might help narrow down whether Opera is intercepting the side buttons rather than the mouse software itself.
Seems like by default registered Opera application is a wrong one, i.e. Logitech is not recognizing it. I deleted it from Logitech Options and added once again (selecting Opera.exe from the installation folder). Registered custom keys on it and all works fine!