There would be no reason to. If DPRK fired a nuclear missile at DPRK unilaterally it would immediately be thrown under a bridge by China because it would be impossible for the Chinese regime to provide cover anymore.
Here's some silly fanfic about a possible lead up to a war with China:
1) DPRK continues to act provactively, to the extent the west decided to take action
2) the Us and its allies introduce a measure in UNSC to give the regime a timeline to disarm or else face surgical strikes and presumptive regime change
3) China and America's new, err, BFF veto; anxious about the north being western backed and possibilities of chaos destabilising the region.
4) Trump and Abe say the UN should stfu, and start bombing (conventionally) north korea, while funding ground operations.
5) the Kim regime collapses or flees, but the country devolves into a messy proxy war between America and China/Russia.
6) the trump government announces an explicit anti-china policy (China is a threat to world peace), and begin openly supporting causes like Taiwanese and Tibetan independence. White papers released lauding regime change.
7) all other nations in the region start getting lobbied by the two sides. Most importantly from a nuclear side, is Pakistan (China ally) and India (America ally).
now a further esculation is more hard to figure out, because trump has upset the formerly predictable notion that NATO would swing behind America and Russia against - but they could get drawn in in some combination.
9) flashpoints get worse and worse, and diplomatic avenues are completely suspended. Places like Nepal, Sri Lanka, Korea, South East Asia are all filled with low level conflict, governmental collapse and coups emanating from one side or the other.
10) Taiwan announces it will declare unilateral independence, and that America has promised it military backing. Beijing announces for its part, that they will view this is an explicit act of war.
11)
12) nukes
Obviously there are some hard reaches of logic: although it isn't impossible to get the Iran-Saudi-Israel conflict in; the major risk of nuclear apocalypse - a large interchange of nukes between Russia and NATO is less likely a prospect. Still, even if it did happen (and I doubt it, I'm not ag) it would be horrific and a further blow against the idiocy of nationalism.
