Is the idea then that there is no way, even in principle, to have S and M be isolated from the wider environment such that there is no record of prior measurements or is it theoretically possible to construct such an experiment and test the two different predictions?

So $\lvert| \frac{1}{2}, {\frac{1}{2}}^{*}$ and $-\lvert| \frac{1}{2}, {\frac{1}{2}}^{*}$ should be considered separate, orthogonal states because they correspond to different worlds? Don't you then have to add some postulate to specify when a new world is created?

I am sorry but I cannot edit my comment. I meant to say $\lvert \frac{1}{2}, {\frac{1}{2}}^{*} \rangle$ and $\lvert -\frac{1}{2}, {\frac{1}{2}}^{*} \rangle$ would be the same. The picture I have in my head for the measuring device is of a needle pointing either to $\frac{1}{2}$ or $-\frac{1}{2}$, depending on what it has most recently measured, leaving no trace of any prior measurements.

An important difference between your calculation and mine seems to be that you assume that the state of the measuring device after the second measurement depends on both measurement outcomes, whereas I assumed it only depends on the most recent measurement so, for instance, $\lvert \frac{1}{2}, {\frac{1}{2}}^{*} \rangle$ and $\lvert \frac{1}{2}, {-\frac{1}{2}}^{*} \rangle$ would be the same state under my assumption. Am I right in thinking that's a key difference? If so, could you justify your assumption?