A sufficiently skeptical position is completely immune to criticism, or to any other form of argument. I don't see what anyone could hope to do about that, beyond not bothering arguing with people who profess such extreme skepticism.
(I am reminded of a little fable I think I saw in an old OB post. Human space travellers encounter an alien planet whose inhabitants have adopted an anti-inductive principle, with the unsurprising result that pretty much everything they do is miserably unsuccessful. The humans ask them "So why do you keep on doing this?" and they say "Well, it's never worked for us before...")
Which means that anti-scepticism is a position taken on faith in the religious sense. It is, after all, the anti-sceptic who claims something can be known.
What I'm looking for is an argument that starts from no assumptions whatsoever but the self-evident, that gets to a justifiable probability theory. That would get around arguments such as the Evil Demon argument.
I've raised arguments for philosophical scepticism before, which have mostly been argued against in a Popper-esque manner of arguing that even if we don't know anything with certainty, we can have legitimate knowledge on probabilities.
The problem with this, however, is how you answer a sceptic about the notion of probability having a correlation with reality. Probability depends upon axioms of probability- how are said axioms to be justified? It can't be by definition, or it has no correlation to reality.