Camping on an island with no man made structure sounds nice. However, the reality is that you will have hot sun beating down on you during the day and perhaps some rain and a long 12 hour night. Having camped in the tropics I can tell you that I'd much prefer to do it under a roof. With the possibility of rain it is risky to sleep without a fly and it's sweltering hot with it on. During the day it's wonderful to have a source of shade and you can't sit under the coconut tree.
What I would do is sleep without a tent, laying my Thermarest mattress on one of the benches. As the area around the shelter is cleared out I would expect it not too be too buggy and just light some mozzie coils, apply bug repellent and use a top sheet. One can also safely pitch a tent without the rain fly.
As you can see there is plenty of space to spread your stuff out on. There is a cooking area and a toilet (clean, no stinky smell).
There are plenty of coconuts available for drinking, you can ask Mr. Obak to prepare some for you. (He was kind enough to send me on my journey back to Koror with several.) There is nothing more refreshing than the juice of a fresh coconut.
