Have you tried authenticating by passing the oauth token as a header, not query param, as stated in the documentation?
curl -H 'Accept: application/vnd.twitchtv.v2+json' -H 'Authorization: OAuth <access_token>' -d "channel[status]=Playing+cool+new+game!&channel[game]=Diablo" -X PUT https://api.twitch.tv/kraken/channels/test_user1
You can use headers{} field in jQuery to send custom headers.