BuddyPress.org

Users are created fine but the member links to their name not their username, so clicking on the member name redirects to the home page, not the user profile.

That sounds like it has to do with how the real name/display name and username are stored in xprofile, wp_users and wp_usermeta, which is a mess in default BP. My custom synchronization code would probably fix that, if I can figure out where to add it in the plugin.

Do the Facebook logins even have a username – lowercase, one word? Would you have to generate one, from the full name? Doesn’t the plugin do that?

Apologies if I’m off base. Haven’t even tested myself yet.