playing this game with a single die roll and all possible values of k recovers the Sleeping Beauty problem

I have no idea what this sentence is supposed to mean.

I've done the same work of formalization for PSB that I did on AlephNeil's revival question and the joint distribution table does have the same structure, in particular I agree that 1/21 is the right answer in PSB. So I agree that this could be a promising start - but it's unclear (and was also unclear with AlephNeil's question) how we get from there to SB.

Given that structure for the joint probability distribution I get results which agree with neq1's answer of 2/22 in the variant described here and don't know where you're getting 1/21 from.

I would like to strongly recommend that we settle this by writing out the full joint probability distribution table (for N more manageable than 20), this isn't so hard to do as long as we're playing with discrete variables with not too many values.