French Polynesia is one of the toughest awards to book, because service is limited. There aren’t a lot of options for getting there, even if you’re willing to connect. The major carrier is Air Tahiti Nui, which serves Los Angeles – Papeete. Air France also operates that route three times a week. United flies three days a week from San Francisco and has good award space in coach. Hawaiian flies Saturday service from Honolulu.
Whether or not business class awards are possible is a yes/no proposition (usually no). There’s not a lot of tricks to make it happen. Although right now Air Tahiti Nui which is an American Airlines and Delta partner is making great business class award space available for two passengers between Los Angeles and Papeete, Tahiti for October through December.