No, am I not Thirding. Thirders consider Beauty's wake-ups on Monday and Tuesday to be "events". That's not what I'm doing here.

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We'll need to be a bit more specific about your Alex & Beauty set-up: I hope you don't mind if we specify how Alex's waking day is chosen. Let's say it's chosen to be Monday if a fair die roll turns out Odd, or Tuesday if Even.

Now, you must agree that there are precisely two independent sources of objective chance in your experiment: the coin and the die roll. There are four equiprobable outcomes, of course:

Pr(Heads,Odd) =
Pr(Heads,Even) = $\frac{1}{4}$
Pr(Tails,Odd) = $\frac{1}{4}$
Pr(Tails,Even) = $\frac{1}{4}$

When Beauty sees Alex awake, she must discount the outcome (Heads,Even) from the sample space, leaving Pr(H,O) = Pr(T,O) = Pr(T,E) = $\frac{1}{3}$ . I really hope that everyone should agree that those are the correct posterior objective chances.

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This analysis says nothing about Beauty's subjective credence for Heads, nor does it attempt to justify any philosophical issues arising. We can have that philosophical debate later. But let's ground the discussion in something that's not open to interpretation by at least agreeing on the objective chances.