Override site's registration restriction on accepting invite
This is a bit hacky, but it could be done with the following:


  
  my_website.com/admin/config/people/accounts
  
  
  Who can register accounts? - Visitors
  
  Install https://www.drupal.org/project/redirect and add redirect /user/register to a basic page which says something like "Sorry, to register you need an invite. Please contact us to receive an invite."
  


Invited people can now log in to site but there is no way for people to access the user register page, as that page will redirect.