I want to have the profile menu item nested inside the Logout menu, which I have made by duplicating the Login module, setting the Title to "Logout" and making it only visible to registered users and setting the Login module to display to guest only. I have set "Show Profile Link" to "Yes" but I don't see a profile link. I created a "Profile" page in menu manager, so I can show the Profile link in the navbar to registered users, but I would like that link to appear inside the Logout menu instead.

You can not set a link into the login/logout because it is the basic joomla login module which is published on module position menu