One huge concern the Chinese have is that they don't want hundreds of thousands of refugees fleeing a collapsing DPRK.
But now China is perhaps the ROK's largest trading partner. And while Comrade Kim's shenanigans can provide a distraction of U.S. attention from Chinese perfidy in the East and South China seas, they also provide a pretext for South Korea and Japan to seek offensive, perhaps nuclear, weapons options of their own--as well as anti-ballistic missile systems that may get hooked up with U.S. systems.
For my money, the Chinese should offer a deal: They take out Kim and let Korea reunify--at the cost of the withdrawal of U.S. forces from Korea.
I tend to think that the Chinese would like to assassinate Kim Jong Un and install a stooge after a coup (a la the Soviets in Afghanistan in 1979). The problem is: Who can they recruit and how does the Korean People's Army react?
Based on China's responses so far, it is unacceptable to China.