While I encourage the idea of theists (or indeed atheists) criticizing the poor arguments of their own side, this particular effort is quite weak. Not because the argument is in fact strong, but because the criticism fails to expose its true weakness.
The argument in question is: in order to know that there is no God, you would have to know everything, i.e. be omniscient yourself.
The true flaw in this argument is this: it mistakes a universal claim for an existential claim. To know with certainty that no black swan exists, you would have to examine every swan in the universe; the negation of the existential claim "there exists a black swan" is the universal claim "all swans are not black". But no statement concerning an omni-attributed god can be simply existential; the claim "there exists an omni-present God" also means "for all places, God is present there" – a universal claim nested inside the existential one.
Accordingly, one does not need to survey the whole of spacetime to disprove an omni-present God; it suffices to show one single place which lacks a God (according to the theist's definition of God). Similarly, an omni-benevolent God is refuted by a single unnecessary evil.
Beaumont's argument is that we can refute the existential claim with logic, which is fine as far as it goes, but notice that this results in devaluing the significance of the evidence against God. Beaumont is obviously a Rationalist in the sense I attack in Against Rationalism; he apparently hasn't considered that we can have strong inductive evidence against the nested universal claim, even if we have no deductive proof of impossibility.