TL;DR: GW crowds are barely noticeable on the Kumano Kodo, it's a great activity to avoid GW-related overcrowding. Japanese people barely walk the Kumano Kodo.
Since my Japanese friend could only get enough time off during GW, we decided to walk part of the Nakahechi (Ayukawa to Hongu over three days). We were worried that it would be super busy, but this worry turned out to be baseless.
GW has almost no effect on the Kumano Kodo because Japanese people barely walk it to begin with. Few Japanese know about it and even regular hikers don't seem that interested, they usually favour actual peaks. This surprised me a lot, but all the guesthouse owners said the same - they are surprised when they get a Japanese visitor as it's so rare.
There were quite a lot of people at Hongu Taisha and the Kuroshio train back out of Kii-Tanabe was also fully booked, so people do go and visit these famous spots, but most don't seem to walk the entire trail. It seems that people mostly know about the KK because of the dual pilgrimage with the Camino, and a vast majority of people come to walk it to get their certificate.
We did struggle a bit with accommodation since we only booked about two months out, but we managed to find guesthouses in Kurisugawa, Nonaka and Hongu without too much issue. We were also worried about the bus back to Kii-Tanabe being crowded, and while most of the seating was occupied, it wasn't full beyond that (mid-afternoon).
If anyone's interested in my specific walking experience I'm happy to expand the post, but I mostly came to relate the trail conditions during GW in case anyone in coming years is worried about this. Just make sure to reserve the train in and out a couple of days in advance, since the Kuroshio is reserved seating only. Beyond that, GW seems like an ideal time to do the hike. We had great weather too.