Registered users only but allow registrations

I have a website that is not yet ready for public consumption, but I would like a group of users, who I will invite by email, to be able to register to essentially beta test the site.
I have a plugin set up that allows me to approve anyone who attempts to sign up so having the registration page public isn't a problem.
So what I want is for any user coming to the site to be presented with the login page (Registered Users plugin does this) but for people to be able to click the 'Register' link and be able to do so. With the Reigstered Users plugin they are just redirected back to the login page.
Would Multisite Privacy plugin enable me to do this?
Do you have any other suggestions for how this could be achieved?