I went and finished the process manually, and here is the resulting climate map (including the climatologically relevant oceanic currents, for easy reference):
Köppen Climates.jpg
For purposes of comparison, here is a prior version of the climate map I did back during the summer:
ERMap.png
Overall, I think it's a pretty big improvement over the previous map, which was partly done based on Pixie's tutorial and mostly done by just looking at Earth's Köppen climates on similar latitudes, or with making "educated guesses".
Although it has to be said that the final climate map shown here has been heavily "corrected" manually at places. It turned out that I screwed up with the precipitations, and Mediterranean climates were practically nonexistant, so I had to put them in manually. Also, I narrowed down the temperature combinations into 20 (from the maximum 100 possibilities) after merging the temperature maps, removing any weird "fringe areas". I didn't bother with refining the merged precipitation map, since I essentially did this manually by first selecting a temperature category (from the merged temperature map) and then going through the different precipitation areas within said temperature category. Theoretically, this would still result in 720 possible combinations, though not nearly all actually turned up.
Still, it took a long time. In hindsight, it might've been better to reduce the temperature categories still more, theoretically you only need 10 to do the different climates (I think). Merging the precipitation maps also resulted in a lot of "fringe areas" with weird combos, especially in mountainous areas. If I were to do this manually again, I'd consider maybe refining the merged precipitation map a little before defining the climates. In any case, what this tells to anyone doing this process via script: don't expect a perfect result, there will probably be a lot of weird fringe areas that will require manual corrections. Also, you need to be pretty careful when doing the precipitation maps, since at least I found it's pretty easy to rush them, and the accuracy of the climate map suffered as a result.
What I also found out in the process is that for purposes of fantasy mapping it might actually be worth it to expand the Köppen classification to include dry polar climates (in my map, I added ES and EW, steppe tundra and polar desert). Present day Earth doesn't really have these climates (as far as I know), but during the Ice Ages they would have been pretty widespread. I think the reason why dry polar climates are marginal nowadays is because all glaciated landmasses are practically islands, with no major non-glaciated areas. This means that tundra climates only occur in maritime areas, without extending far inland.
That said, it's common in fantasy maps to include landmasses that are only partially glaciated (essentially similar circumstances to Ice Age Eurasia). In these kinds of circumstances, you'd probably have an Ice Cap climate (EF) surrounded by polar desert (EW), transitioning into steppe tundra (ES).
EDIT:
It's also worth mentioning that I didn't quite realize until comparing with Earth's Köppen climates that the influence of the Winter Westerlies extends all the way to the Hindu Kush and Pamir mountains, resulting in "dry summer" climate types. It might be worth it to add a footnote about this into the precipitation section. At least I seriously underestimated how far inland the effect of the Winter Westerlies extends.