I've got a OWE coming up with flights to and from Easter Island included for January, so there's definitely no problems in getting that added (Easter Island is considered part of South America, as opposed to SW Pacific).

If you're wanting to go directly from Easter Island to Tahiti and on to New Zealand it's probably worth looking at a Global Explorer ticket. That lets you use the Qantas codeshares out of Tahiti and other places around the Pacific, so gets you there directly.

Otherwise you'll need to buy or use an award booking to bridge the gap between Tahiti and New Zealand.