No; you are asking her two different questions, so it is correct for frequentism to give different answers to the different questions.
Of course. But the two questions are the same outside of anthropic situations; they are two extensions of the underdefined "how often was I right?" Or, if you prefer, the frequentist answer in anthropic situations is dependent on the exact question asked, showing that "anthropic probability" is not a well defined concept.
A friend referred me to another paper on the Sleeping Beauty problem. It comes down on the side of the halfers.
I didn't have the patience to finish it, because I think SB is a pointless argument about what "belief" means. If, instead of asking Sleeping Beauty about her "subjective probability", you asked her to place a bet, or take some action, everyone could agree what the best answer was. That it perplexes people is a sign that they're talking non-sense, using words without agreeing on their meanings.
But, we can make it more obvious what the argument is about by using a trick that works with the Monty Hall problem: Add more doors. By doors I mean days.
The Monty Hall Sleeping Beauty Problem is then:
The halfer position implies that she should still say 1/2 in this scenario.
Does stating it this way make it clearer what the argument is about?