First, an obligatory reminder that the original post is not an injunction against talking politics, it is against politicizing a non-political discussion.

Do you think that the very same idea of rationality can be the subject of existential risk?

Second, let's imagine the future where the fundamental/radical/militant Islam won. Specifically, the Salafi version of Sunni Islam. Unconditionally. Everyone is muslim, no other religion is allowed. Everyone not originally muslim is either converted or killed. What's the worst that can happen to "rationality" in this case?

Clearly this would not affect instrumental rationality much, as striving to succeed is what people do. Also clearly, anything associated with Jews would be purged. This forum would be one of many casualties, but probably would not even register on the scale of destruction. Now, if you look back into history, this happened before, multiple times in various places. Only the dominant religion and the outgroup are different. Almost every time it was a setback, but only a temporary one. Science survived and eventually prospered. The dominant religious movement mellowed, then eventually splintered and crumbled, freeing the epistemic rationality to develop once more. I see no reason that this would be different the next time around.

What do Neoreactionaries think of the Islamic State?

I would imagine they would reply with their usual metaphors, about the Cathedral, Cthulhu's swimming habits and so on.