I am having the hardest time finding shoes for my son. When he could wear the smallest sizes we had a few more options, because there are lots of flexible sole options. He is now in a size 10 and I'm having no luck. His toe box is very wide (Keens aren't wide enough) and his instep is high. I insist on flexible soles. The poor kid is down to one very worn pair of Tsukihoshi shoes, that although he says they are comfortable, do seem to be a bit ill fitting on the instep. Please throw out your ideas.
People may know of some brands that I don't, but I would move on to more traditional shoes at this point. DS has similar feet and when he was that size we started doing Stride Rite shoes. I don't like them for little ones but I'm assuming your DS isn't a toddler anymore.
This may not qualify, but I can bend the soles with my hand. My DS has difficult feet to fit and a pair of wide shoes from New Balance (I bet my mom got them at Kohl's) are what fit him best now (he's in a 9 or 10). You could check them out - they aren't super soft soles - not as flexible as our Robeez Tredz were, for instance, but they aren't totally totally hard.