Anime order seems fine to me. Though, moving Nayuki’s position seems a bit odd, since you can’t really call it the anime order anymore (even if it’s mostly that way).
It would seem a bit more “smooth” to follow some other metric, like the realism of each route (since some use less abstract elements than others), or the amount of discussion you expect each route to offer in the podcast. For example, going from most realistic to least, I’d say something similar to this:
Nayuki, Shiori, Ayu, Mai, Makoto.
Nayuki being closer to the discussion of the common route makes sense (so does just merging the two). For narrative purposes I think Ayu should be last though. Obviously, the example ordering above does not meet that.
Given that, and considering that it’s more or less inconsequential (i.e., the ordering doesn’t really matter), the slightly-modified anime order probably offers the most familiarity for those who have only watched the anime.
As for Flutterz’s recommended order (Ayu, Nayuki, Makoto, Shiori, Mai)… it seems to be taking the approach of explaining the most things to the least? I’m not too sure about the reasoning behind that order. Though I don’t expect a response to arrive soon enough to be practical here, I’ve sent a message along to see if they would share their rationale for the order. I think it would be interesting to hear their perspective on it.
TL;DR: That mostly-anime order is good enough and it’s unlikely that any further effort will yield a measurably better experience for bookclub participants.