You can modify the launcher so that it runs Opera automatically with the additional options. I tested by adding a third launch action. Now, I can right-click the Opera icon and select New Window with "O". You could also make this the default behavior. Here's how I did it.
In Terminal, I edited the desktop info for Opera.
sudo nano /usr/share/applications/opera.desktop
Actions=NewWindow;NewPrivateWindow and then defined the action by adding the following to the bottom of the file:
[Desktop Action NewWindowWithO]
Name=New Window with "O"
Exec=opera --show-opera-menu --new-window
Save the changes. You should now be able to right-click the Opera icon in dock and select the new action.
If you want the default action (left-click) to open with the "O" menu, then you can add
--show-opera-menu to the
TryExec fields in the
[Desktop Entry] section.
I'm not sure if this file will be overwritten if you install a newer version of Opera, so you may want to delete or revert it to it's original before you update.