I think you have it backwards. Firing missiles over Japan is a form of shakedown. What the Norks want is money. As long as they can keep the Chinese subsidizing them they can stay in power. They keep the cash flowing by keeping the Chinese afraid of what will happen if their regime collapses, or if it provokes a conflict with the US and its allies in the region.
” Firing missiles over Japan is a form of shakedown.”
True, but that is North Korea’s motivation.
For China I’d think they might still be concerned/angry with Japan and not anxious to push the Japanese to further rearm (or heaven forbid actually assemble some of there nuclear weapons).
Also, if China were to see a US action as inevitable then they are faced with an unwelcome set of alternatives: 1) Oppose the US militarily, 2) Drop support for NK, 3) Pressure NK into behaving. Avoid #1, Implement #2 early to maintain credibility, Pursue #3 (what’s to lose?).