But the information gained is what we sometimes call 'indexical' information - information about where, when, or who you are. When you wake up, the thing you learn is that you are now inside the experiment. That seems like a pretty important new thing to know.

Exactly! To quote Bostrom

On this reasoning, it would seem, one could similarly argue that when Beauty awakes on Monday (but before she is informed that it is Monday) she likewise gets relevant evidence – centered evidence – about the future: namely that she is now in it.

Incidentally, I had a question on that paper, and now seems as good a time as any to bring it up. To quote the second-to-last paragraph (this will make no sense unless you've read it)

It is interesting that in Beauty and the Bookie, Beauty’s betting odds should deviate from her credence assignment even though the bet that might be placed on Tuesday would not result in any money switching hands. In a sense, the bet that Beauty and the bookie would agree to on Tuesday is void. Nevertheless, it is essential that this bet is included in the example. The bookie is unable to pursue the policy of only offering bets on Monday since he does not know which day it is when he wakes up. If we changed the example so that the bookie knew that is was Monday immediately upon awakening, then Beauty and the bookie would no longer have the same relevant information, and the Dutch book argument would fail. If instead we changed the example so that Beauty as well as the bookie knew that it was Monday immediately upon awakening, then Beauty’s credence in HEADS & MONDAY would be 1/2 throughout Monday, so again she would avoid a Dutch book.

I didn't really get how this would work. If she doesn't lose anything on the second bet, then that's effectively not a bet. How can losing nothing be part of her expected loss calculations?