Hi Rael,
To add a new sitelink, you need to retrieve the existing CampaignExtensionSetting, add the new SitelinkFeedItem to it and update the CampaignExtensionSetting using the SET operation. ADD operation won't work in this case, since you already have added some sitelinks to the campaign.
Cheers,
Anash P. Oommen,
AdWords API Advisor.