Is there a way to allow sign-up only for Twitter integration?

Problem with making the site invite-only is that the SignUp button doesn’t show up in the login page, and I want uninvited users to still be able to sign up but only through an associated account.

Basically, “You can sign up with Twitter/LinkedIn/Google. Otherwise, if you don’t have any of these, you have to be invited.”

(I realize I can override the Sign Up page to remove the input fields and “Create your account” button on the left side, and only show the associated accounts buttons, but I wonder if there’s a way to avoid that.)