The most common solution to provide a Facebook Login and re-use Facebook account details in your own login system is, to use a Laravel library like Sentry, which supports OAuth2(http://oauth.net/2/).
Because you are forcing a Sentry::login($user, false); for the user without a password, you might want to generate a password field to have a regular non-facebook login. After a successful login to your website with OAuth2 against FB, you simply store the client-secret, log the user into your system, fetch the FB infos you need and send an email with a generated PW for direct login to your application. Email: "Thank you for registering to this website via Facebook. Your PW is ....".
As pointed out by @har2vey, it's the same functionality as provided by the "resend my password" feature, but directly send after login via OAuth and not on user request.
A Sentry example is already on SO: https://stackoverflow.com/a/19511907/1163786
Sidenote: It's unfair towards those who take the time to answer, if you simply delete your questions. I'm refering to switch-profiles-on-the-client-and-server. The answer could be useful to others aswell.