The clips endpoint probably requires authentication using the session cookie + API token from the website.
Judging from my own experiments the API token used by the official site is useless for API requests without the session cookie and vice versa. Twitch’s on-site login requests a special 'openid' scope which you can’t use with the official OAuth flow (IIRC you get an 'invalid_scope' error). Note that this information is based on my own recollection of results playing around with the login session. It’s not official information, nor do I recommend doing this 
If you’re just going to use it for yourself you can always “hijack” your own login session (Twitch-Api-Token header, persistent and api_token cookies) and copy+paste it into your application. Login credentials are valid for ~30 days.
If you’re building a service intended for use by others you’re probably out of luck. There’s not much you can do besides waiting for the clips endpoint to become part of the official API.