CB is designed with open registration. It has no invite only registration process. You're welcome to check Joomla.org extension directory to see if something is available or the easiest route is to add a new CB field from Field Management which uses Perl Regular Expression (REGEX) validation to validate against a single word (meaning if they don't supply that word they will be denied registration). You could then email that word to users of choice for them to come register. Example of validation as follows.

/IAMAVET/

If a user did not supply exactly IAMAVET they'd receive a validation error when trying to register assuming this field is set to display on registration and is set to be required.

Please navigate to CB configuration followed by registration tab and ensure CB registration is independent of Joomlas. Once done navigate to Global Configuration and disable Joomla registration. Then check your CB Login module parameters to ensure you did not disable the register link.

This sounds weird, but I can't find the registration (Joomla) button in my global config???

Don't understand what you're looking for. There is no button in global configuration. It's a parameter you set from Yes to No within global configuration as follows.

Site > Global Configuration > System > User Settings > Allow User Registration > No

Once the above is done please perform the below.

Components > Community Builder > Configuration > Registration > Allow User Registration > yes, independently of global site setting

Now perform the below to ensure the login link is present on your module.

Extensions > Module Manager > - Select Type - (FILTER) > mod_cblogin > Edit ALL (1 by 1) > Parameters > Create account > Show

What's up with that?

You've unpublished the default template. Please navigate to plugin management and ensure is published and public.