Two ways this goes:
1. DPRK actually tries to fire something. It lands and destroys stuff and kills people. They get flattened into a pancake.
2. DPRK tries to fire something and it fizzles and/or is intercepted because their technology is fifty years out of date. They lose face, we laugh at them. They promise to be good, retreat back into their shell for another ten years, and try again.
3. DPRK fires something, kills a bunch of people in SK. The nutty liberal anti-war crowd in the US takes this as an excuse to argue that we shouldn't have been in the region anyway since our presence is what's causing all the strife, and we pull out in order to appease them. China gets the hegemony they've been building towards for the last 50 years.
I don't think that's likely to happen, and I believe that threading the needle between just enough to make that argument work in the US, and enough that we'll just flatten NK would be nearly impossible, but scenario 3 is exactly what the Chinese want to happen. I'll also point out that us flattening NK and then withdrawing from the region works for them too. The only actual lose scenario for China is one in which the US commits to *more* military/economic presence in this theater, which is unlikely to happen. Doubly so given we wont ever actually do anything directly to China unless they overtly do something directly to us. They're more than willing to let NK do something crazy, allow us to bomb NK in retaliation, but then step in and insist that we can't actually overthrow the NK regime with threat of getting directly involved if we do (more or less exactly what they did during the Korean war). If NK "wins", China wins. If NK loses, China maintains the status quo.