This is precisely what I meant when I mentioned the empirical side information detector. The "informational point of view of Dumbledore" is "whatever-it-is that keeps histories consistent," and the indistinguishability only has to come into play in the local context of whenever Dumbledore uses the time turner. In the way I've envisioned it to work, Dumbledore can only use your algorithm to detect leaked information or side-information that was available to him which he might not be aware of.