Another argument for halfing
What about this much simpler argument for halfing:
As usual, Sleeping Beauty wakes up on Monday, knowing that she will have an indistinguishable waking experience on Tuesday iff a certain fair coin has landed tails. Thirders say her credence in the coin landing heads should be 1/3; halfer say it should be 1/2.
Now suppose before falling asleep each day, Beauty manages to write down her present credence in heads on a small piece of paper. Since that credence was 1/2 on Sunday evening, she now (on Monday) finds a note saying "1/2".
If Beauty is a halfer, she already knew upon awakening that her credence in heads the evening before was 1/2: it will have been 1/2 no matter if it is Monday or Tuesday, and no matter if the coin landed heads or tails. So the note tells her nothing new about the coin toss, and she should stick to P(H) = 1/2.
But what if she's a thirder? Let x be her new credence in heads after reading the note on Monday. There are two possibilities: either x equals 1/2 or it does not.
Suppose x equals 1/2, i.e. Beauty's credence in heads after reading the note is 1/2. Assume she knows that she will receive no further relevant information that day. Then she knows that she will find another note saying "1/2" the following day. So even before reading the note, she could have figured out that it will say "1/2": if it is Monday, it will say "1/2" due to the known Sunday credence, and if it is Tuesday, the note will say "1/2" due to the effect of the previous note saying "1/2" on Monday. So the note tells her nothing about Monday vs Tuesday, or heads vs tails, and her credence in heads after reading the note should equal her prior credence: 1/3. This contradicts the supposition that her credence in heads (= x) rises to 1/2.
Suppose x does not equal 1/2. Assuming she knows that she will receive no further relevant information that day, she can conclude that the following day she will find a note reading "x", and not "1/2". So even before looking at the note, she could have figured out that it will say "1/2" on Monday and "x" on Tuesday. Hence upon finding that the note says "1/2", she should infer that it is Monday. Now on the thirder account, P(H|Mon) = 1/2. And since the note contains no information relevant to H & Mon versus T & Mon, Beauty's credence in heads after reading the note should therefore be 1/2 -- contradicting our supposition.
If all this is correct, then if Sleeping Beauty is a thirder, there is no rational way she can incorporate information about her credences from the evening before. This doesn't only show that she is irrational in the particular scenario I have described: being incapable to rationally incorporate new information is irrational even if there's merely a positive chance that the information will come up.
This is the coolest argument for halfing I've ever seen.
I think that under "If Beauty is a halfer", the bit about "it will have been 1/2 no matter if it is Monday or Tuesday" is too quick, as these probabilities pertain to the original sleeping beauty situation, not your modified situation. Prima facie, there isn't an obvious incoherence in Beauty being a halfer in the original situation, but rationally assigning x=2/3 in this situation. Assuming that she could know in advance that she'll assign x=2/3 in this situation, then finding 1/2 on the note will tell her that today is Monday. P(heads|Monday)=2/3 for halfers, so assigning x=2/3 is rational. Of course there isn't an obvious incoherence in her assigning x=1/2 in this situation either. A case of rational underdetermination?
Furthermore, once we note the possibiity of this sort of rational underdetermination, a key element of your reductio argument against thirding is undermined. A key premise of the reductio argument is that if Beauty rationally forms a credence of x in a situation, then she knows that she will always form a credence of x in such a situation. But if rational underdetermination is possible in these situations, then this premise may be false.
Of course lots of tricky issues immediately arise (not least that the falsity of the premise would itself raise a problem in the initial argument for rational underdetermination). But I suspect that getting to the bottom of this sort of issue may be crucial. The choice between reductio and underdetermination is interestingly reminiscent of the difference between liar sentences and truth-teller sentences. My guess is that there is more to this analogy.