By right, it shouldn't allow you autopost to your Facebook user profile page if your stream privacy not set to everyone, because Facebook unable to retrieve those stream data if privacy set to registered user only can access.

But I couldn't find "Everyone" here
see the screen shot

Since you enabled lockdown mode which means only registered users are able to view contents, it doesn't make sense to have "Everyone" in the privacy list. Hence, it also doesn't make sense for auto posting because there is no way for Facebook to crawl the contents on your site

If you want such functionality on the site, you should turn off lockdown mode.