If she was told she would be woken up 3^n times if n is even, 0 times if n is odd, then it seems obvious enough that when asked upon being woken up what she thought the probability that n is even, she would rationally and correctly say 100%. And that this would make sense. So Why wouldn't it make sense if the answer is some number other than 100%?

What she would do differently is bet on things she cared about based on the odds. Like "would you rather your relatives are given $5 if the number of coin flips is odd or $3 if the number of coinflips are even?" The answer for a rational beauty would depend on the probability that the number of coin flips is even.