CCP wins the PR lottery!
Doubt China would go to war over something like this. North Korea would go begging forgiveness, China would say some harsh words and tell them to cool it on the missile program and it'd probably all blow over after a month or so.
That is my bet too. And the PRC ladership will milk this of course for all it is worth.
They will face an outcry at home of course. But then: The CCP just loves it when they can feed their population something to get riled up about a foreign country on the news. Ideally about Japan. Second best if about the US. But really: Any country will do as a scapegoat. I still remember how the news were paced here in China not too many years ago. It was always: First segment had Head of State today did this or that, shakes hands with what's-his-name, cementing how important China's Main Man is in the world. Second Statement: Report on a minor problem issue in China, ending with how local officals deal with it competently and optimistic. End segment: Gloom and doom in foreign country.
Things have changed meanwhile - but that only means that the scenario the OP describes can be the first segment today instead of being the last.
Chinese citizens will then blog and write and get all riled up online. Maybe some even stage a demonstration here or there. All with the blessing of the state - unless it gets too out of hand. Then discreet censorship steps in online. And demonstrations on the street ... they are partly staffed and organized by the state anyhow. Domestic mission accomplished nonetheless: China's public is distracted nicely for a couple of weeks from domestic issues.
At the same time, the CCP leaderhip will capitalize internationally on the issue twofold.
First: Get all the condolences coming in from everywhere, showing how important China is as the president of the US, of Russia, of India, of Germany and all these other unimportant nations express their sorry. Almost like the good old days of the Emperors when all the foreign barbarians came hat in hand to the Forbidden City to grovel before the throne. Japan sending condolences will probably not be mentioned too much. But Singapore, Hong Kong and Taiwan definitely will be featured big, conjuring some kind of Chinese unity out of thin air.
Second: Acting all super statesman like and magnanimous by not starting a war or crisis over this. Acting all solemn and level headed, the cooler heads and voice of reason during all the international outrage. Then waiting for the praise from the foreign press coming in pointing out how the US went to war in Iraq over two levelled skyscrapers while China remains calm over a shot down airliner ... even though it could easily roll into and over North Korea.
Demands of compensation for the family of the deceased may be a tricky issue. China's blogospehere and internet forums will demand these. But my bet is the CCP will again play the magnanimous card and generously compensate the families directly - all properly reported on Chinese state tv of course. At the same time appearing generous towards North Korea, the pauper who does not have anything to give to begin with, on an international level. And again raking it in domestically at the same time by showing "See: When disaster strikes, the CCP steps in". In the old days (when I still occasionally watched Chinese tv) this would have been segment two of the news. Abjusted to segment one nowadays. See? Progress!
I am with GFW. I give it about a month of media exploitation by the CCP.