It is, after all, the anti-sceptic who claims something can be known.
The sceptic also claims that something can be known: that nothing can be known.
What I'm looking for is an argument that starts from no assumptions whatsoever but the self-evident
Any conclusion can be reached by choosing the right "self-evident" assumptions; any conclusion can be denied by denying the self-evidence. It is self-evident to John C. Wright that God exists and His angels spoke to him in a vision. It had previously been self-evident to him that no such thing was possible. His rational mind pursues the implications of his latter beliefs as relentlessly as it did his previous beliefs. (His blog makes a fascinating case study in taking ideas seriously.)
What will be self-evident to you? Only you can decide that. You would be better off reading diverse books on the foundations of probability (Jaynes, Feller, Savage, de Finetti, Kolmogorov, ...) and seeing whether their arguments convince you, than asking someone to come up with an argument that you will find compelling.
I highly doubt those authors address the idea of scepticism in the sense of showing from first principles that probability is legitimate, in a way that addresses such things as Descarte's Evil Demon Argument. They are discussing how to implement probability, not whether the entire concept is legitimate.
You are denying the validity of self-evidence here, and admittedly there is a problem with properly establishing self-evident ideas. That's part of my problem, admittedly, and what I am trying to bypass somehow.
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.