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I tend to agree with ikalugin. Your looking at way more then 6. Depends on how far away this island paradise of yours is, you may be able to cut the stores people, cooks, janitorial.

But to sail an SSN your going to still need a full reactor watch, crew handling the propulsion and auxiliary machinery. I'm the control room your going to need a navigator, and I believe that in the LA class that you had a helmsman and two planes man. You will need someone at the ballast controls.

The thing is that american naval doctrine is very much crew intensive on ships and subs. That's why the crew size on an American submarine is so large. You may be able to get away with a handful of men on a Russian submarine Because they incorporated far more automation in their sub designs then western subs. If you take a look at the crew of a Russian sub a fair number of that crew is officers.