The likelihood ratio P(E|H)/P(E|~H) is entirely independent of the prior P(H)

In theory, yes, but we're talking about a purported "unswayable Bayesian". If someone strongly believes leprechauns don't exist (low P(H), where H is "leprechauns exist" ), they should strongly expect not to see evidence of leprechauns (low P(E|~H), where E is direct evidence of leprechauns, like finding one in the forest), which suggests a high likelihood ratio P(E|H)/P(E|~H).

I remember Eliezer Yudkowsky referring to typical conversations that go like:

Non-rationalist: "I don't think there will ever be an artificial general intelligence, because my religion says that can't happen."
EY: "So if I showed you one, that means you'd leave your religion?"