that doesn't follow. Where would Harry have gotten the pies if not from Harry+1?

He got them from the breakfast table. Where did he get the idea to get them? Well, he would have seen the pies later on anyway. Just like he would have learned about time turners later on in the day anyway, but a more stable scenario was obtained by learning about them earlier.

The recursion is non-iterative beyond the number of loops actually manifested, however.

I'm not quite sure how to parse this. If you would think about an idea at time T, but don't because future you tells you it won't work out, that means your whole thought process going forward has completely changed. But maybe the thing you start thinking about instead doesn't work out, so someone warns you about the idea at time T+epsilon. And so on. So if you are proposing that Time works by iterating through a number of scenarios until you get to something stable, the situation you've pointed to "requires recursion." (It's worth pointing out that Harry, when he gets his Time Turner, doesn't think this is a likely answer to how Time works.) But perhaps I am not understanding you correctly?

My main objection to the scenario you are proposing, though, is that you are gaining information as a result of some work, but that work is never performed. Try taking your scenario to its logical extreme. You sit in a room with one copy of future-you, and a large composite number N on a sheet of paper. On scrap work hidden from future-you, you write down an integer K. If you are not told that K does not divide N, you check. If it does, you keep track of the factor of N you have found. In any case, you then systematically select a new integer K' to check for divisibility. Once you have a complete factorization, you sit quietly, and at the end of the hour you go back in time. Then, you let past-you know the "keys" for all of the integers that weren't factors. Thus, you must have ended up only trying actual factors. So, you have a slightly more complicated version of Harry's factorization algorithm.

Edit to add: I guess this situation actually still is still an exponential time algorithm, since you still have to consider every possibly integer. But you could do, for example, graph isomorphism testing in polynomial time with a similar method: try constructing a map by first finding what vertex 1 should map to. Future you says "nope, nope, maybe," and on "maybe" you try to figure out where to put vertex 2, given that vertex 1 maps to whatever. If the graphs are not isomorphic, future you will say "no" to every choice for vertex 1, so you have your answer in polynomial time. Otherwise, you will construct the isomorphism in polynomial time.