As I see it, initially (as a prior, before considering that I've been woken up), both Heads and Tails are equally likely, and it is equally likely to be either day.

It's not equally likely to be either day. If I am awake, it's more likely that it's Monday, since that always occurs under heads, and will occur on half of tails awakenings.

I know from the setup that I will be in this subjective state once under Heads and twice under Tails, and they are a priori equally likely. I have no data that can distinguish between the three states of identical subjective state, so my posterior is uniform over them.

Heads and tails are equally likely, a priori, yes. It is equally likely that you will be woken up twice as it is that you will be woken up. Yes. That's true. But we are talking about your state of mind on an awakening. It can't be both Monday and Tuesday. So, what should your subjective probability be? Well, I know it's tails and (Monday or Tuesday) with probability 0.5. I know it's heads and Monday with probability 0.5.