Here’s the problem: what does this have to do with income distribution in a modern capitalist economy? The answer is as follows: nothing. Absolutely nothing. In this island there is no state; people do not cooperate (actually they appear not to have any relationship whatsoever); there is no injustice; there is not even trade. So the thought experiment is worthless.
The point of models being to strip away all that other stuff in order to just discuss the point at issue. In this particular case the point at issue is what is unjust? More specifically, is it just or unjust to force the productive to increase their production in order to supply the unproductive?
there is no injustice
Of course: given that we are trying to explore injustice, we cannot assume injustice in our model. That is petitio principii, a violation of logic let alone economics.