Amanda, wow, that's quite a situation.
I would be very concerned about the rudeness. In your shoes I would try to teach her to play the way other kids play, so she can participate in what they are doing at least for short stretches. And I would do some role playing about how to shift a game in her direction, without being rude.
Sometimes a kid with too few social skills will use academics as a cover. My own DS8 (has Asperger's) has said to me "I'm no good at this social stuff, let's go home and do more science." My opinion is that being with other people is a key life skill and if it's hard, we need to learn it.
You'll have the academics taken care of more easily, since you are homeschooling and can go at her pace. But running out of possible friends on an island sounds worrisome.
DeeDee
