Is the GarageBand App that comes pre-installed on the Mini Retina the full version?

No, it doesn't come preinstalled, but you can download it from the app store for free.

And I'm not sure if it's a full version or if there are options to purchase full functionality.

GarageBand is free on the App Store for all iOS 7 compatible devices; additional GarageBand instruments and sounds are available with an in-app purchase.

http://www.apple.com/ipad-mini/built-in-apps/

Looks to me like you have to buy full functionality

You do have to make an in app purchase if you want ALL the instruments.

How to get all the iWork apps, iPhoto, and iMovie for free on an eligible iPhone or iPad

http://www.imore.com/how-get-all-iwork-apps-iphoto-and-imovie-free-eligible-ipho ne-or-ipadhttp://www.imore.com/how-get-all-iwork-apps-iphoto-and-imovie-free-eligible-ipho ne-or-ipad

 Cheers, Tom 😉