...the details of the experiment do provide context for “today.” But as a random variable, not an explicit value.
You seem to think that “random” variables are special in some way that avoids the problems of indexicals. They are not. When dealing with epistemic probabilities, a “random” variable is any variable whose precise value is not known with complete certainty.
Still, there are ways to avoid using an indexical in a solution. I suggested one in a comment to part 1: use four Beauties, where each is left asleep under a different combination of {Coin,Day}. Three are wakened each day of the experiment. One of those three will be awakened only once during the experiment. They are asked, essentially, “what is the probability that you will be awake only once?” It was agreed that this question is equivalent the original problem.
This is not what I understood you to be proposing. As described here, I would say that this is not the same as the original question, and does not avoid using an indexical. You have simply camouflaged the indexical by omission when you write that “One of those three [who are awake today] will be awakened only once during the experiment.”
You seem to think that “random” variables are special in some way that avoids the problems of indexicals. They are not. When dealing with epistemic probabilities, a “random” variable is any variable whose precise value is not known with complete certainty.
This is not what I understood you to be proposing. As described here, I would say that this is not the same as the original question, and does not avoid using an indexical. You have simply camouflaged the indexical by omission when you write that “One of those three [who are awake today] will be awakened only once during the experiment.”