I have been to Tahiti a few times. The first thing, I'm sure you know this, it is horrendously expensive. However the locals are wonderful, hospitable and charming people and a cycle around the island would be a dream trip. Once away from the capital Papeete I am sure that you will be able to get permission to camp on private land. http://chez.mana.pf/~tahitihostel/en_prix.htm is the only reasonable priced place I have been able to find in Papeete.
Every evening near the waterfront in Papeete and most towns and villages Le Trucks will appear. They serve reasonably priced (for French Polynesia) and plentiful meals and are the only bargains around apart from baguettes which are subsidised and very good. Buy those early in the day as they sell out and shops close for lunch time.
It's worth going to the town quay in Papeete and the marina near the airport to meet up with cruising sailors, they have a lot in common with touring cyclists, come in all nationalities and may be able to help with accommodation or storage (on deck) of a bike box for a short time. Many have folding bikes and may join you for a ride. If you pick up a bottle of gin in Los Angeles or Auckland duty free you will probably be able to sell it to a British yacht or trade it. In fact use up your baggage allowance to the max with tradable goods as anything is worth more here than you paid for it anywhere else in the world. Locals earning Polynesian Franks have a good standard of living but yachties struggle here.
Ferries to Moorea are frequent, I'm not sure about getting to other islands by ferry as I have always been there for work and flown or travelled by yacht. If you get a ride to other islands in a yacht be aware that it will be much harder to get back the same way as the yachts tend to go on downwind and not return to Papeete.
One of my best days cycling ever was around the island of Bora Bora but it is the most over rated of the Society Islands my favourite is Huahine, unfortunately I didn't have a bicycle there but have sailed around it.
If anyone has questions about South Pacific Islands or getting together with yachties please contact me, I may be able to help.