UPDATE 3: Joshua from Gravity Support provided the following info:
At the point in which a user has clicked through to the URL provided by the merge tag {set_password_url} , WordPress itself is handling the process of the password reset. As such, since WordPress provides its own hooks for this, the add-on does not provide any additional hooks for that password reset. You can make use of the action after_password_reset in order to trigger an email to the user.
Sounds like a plan!
Wasn’t aware of that hook! Now I just have to find a way to send that 3rd email only in the case where a user is an admin and only when the admin user is completing the registration/set password process and not as part of any other general password reset.