Saying that they don't occur in the ratio of Bayes' Rule [...]

But I didn't say that. I didn't say anything even slightly like that.

This is at least partly my fault because I was too lazy to write everything out explicitly. Let me do so now; perhaps it will clarify. Suppose X is some long random-looking sequence of n heads and tails.

Odds(Alice cheated : Alice flipped honestly | result was X) = Odds(Alice cheated : Alice flipped honestly) . Odds(result was X | Alice cheated : Alice flipped honestly).

The second factor on the RHS is, switching from my eccentric but hopefully clear notation to actual probability ratios, Pr(result was X | Alice cheated) / Pr(result was X | Alice flipped honestly).

So those two probabilities are the ones you have to look at, not Pr(Alice cheated) and Pr(result was X | Alice flipped honestly). But the latter is what you were comparing when you wrote

it's -always- more likely that Alice cheated than that this particular sequence came up by chance.

which is why I said that was the wrong comparison.