I'd note that "Hirohito abdicates" and "Japan becomes a republic" are two very different premises; the former is much easier to pull off than the latter.
That said, I do think that a republican Japan is certainly doable, but it probably does require an invasion of the mainland (indeed, I suspect it's the probable outcome of a "no-nukes" or "successful Kyujo incident" timeline), simply because the Japanese government wouldn't accept it otherwise. There would be some unrest, but people would get over it fairly quickly. Confidence in the Japanese monarchy was heavily shaken by the end of the war and American occupation, and the US expended quite a bit of effort shoring up the throne to its current status. Society was more or less in turmoil due to all sorts of reasons (including the famine), and people were trying to figure out what the new order should look like, with lots of conflicting ideas. If the US had insisted on republicanism as a prerequisite (something which no major faction of the US government wanted OTL), it would merely mean that that strand of ideas would come to the forefront (and that, as others noted, Japan would probably be significantly further left politically).
I do think that there is a tendency around here to indulge in a somewhat orientalist view of Japanese as a monolithic bloc of fanatics. While "fanatics" does describe certain portions of the military (such as the ones who staged the various coups and attempted coups), it's not really a helpful way of looking at Japan as a whole. The average Japanese in 1945 was starving, impoverished, and likely in mourning for a friend or family-member killed in the war. Most of the major cities have been flattened (even in a no-nukes timeline). If there has been an invasion, it likely has seen even more devastation for the country (and killed off a lot of the more fanatical would-be resisters).