But the question is how the population size relates to health, which you haven't answered here. You have numbers for the number of deleterious mutations, but that doesn't directly give a quantitative relationship between population size and health.
The claim was that the population size of Tristan da Cunha is problematic. I still haven't seen any direct evidence presented of this fact.
You linked to an article about the higher incidence of asthma on that island yourself (above), and the reason why the number of individuals is so high directly traces to the islanders' gene pool.
However, I wouldn't say the population size is unsustainable. They are more likely to be decimated by a random disease or, over generations, by a shared mutation. But the population is not nonviable. The issue is more that, to the modern humanitarian mind, they will have to deal with more ill effects due to the genetic monoculture.