If I could restrict new signups via the :join module and require a user to enter an access code or key to join, is this at all possible without any additional programming? Maybe a feature pre built or addon somewhere?

This would restrict random signups and would permit me to only allow memberships from those who have the access key.

To clarify, you mean a shared secret key that everyone uses to sign up, but isn't actually stored with the account?

I'm pretty sure that's what you meant.
It's not a feature right now, but I like it, I've added to the tracker 0003342: Invite key for joining the site - Composr CMS feature tracker

A workaround would be to set up an httpauth restriction using Apache's .htaccess, on URLs that match that of the join module.

Yes, so basically similar to the recommend feature in that you can only join if a member invites you, but instead you can only join if you have the correct 'passkey', otherwise signup is disabled for normal site visitors.