If you want the Logout button to display in the User Menu that the SCLogin module displays, there's a trick to make it work. You need to go to the Joomla menu manager and create a new menu item as follows:
* Type: External URL
* Title: Whatever you want the menu item to display, e.g. "Logout"
* Link: sclogout - This is a special name that the SCLogin module will use to know you want a logout button.
As for the menu not displaying like your main menu, it likely won't. The User Menu option uses a Bootstrap Dropdown. If you want it to display like some other part of your site, you may need to add some extra CSS to your template to style the items how you want. If you need help, please create a test account so we can login and see the logout view (username: test, password: test is a good example) and let us know a URL we can test on.
We'll help however we can.
Thanks,
Alex