Japanese nuclearization is not an existential issue for China. In fact, this can only help China in the long term: 1) it will make Japan more independent of the US, which increases the chance it becomes a neutral party, 2) it will make it easy to justify accelerating China's own nuclear build up, as the West now has lost its strongest argument against China's nuclear build-up, 3) it'll open the box to sharing nuclear weapons with US opponents.
Obviously, China should still flip a table over it and use it as leverage to 1) punish Japan where it benefits China, 2) excuse its own nuclear build-up above parity with the US, 3) proliferate nuclear weapons to US opponents like Iran and Cuba. Let the US panic & sweat over the proliferation of nuclear weapons around the world, as this can only make it harder for a global imperial power to exert its will via military means, which is what the US does, but not what China does.
The only exception here is if nuclear weapons somehow make it into the hands of Taiwan. THAT does require a military response.
When a nation plans to build nuclear weapons, apart from the need for real life testing, it has to pick a nuclear doctrine:
1- Coercive nuclear arsenal: Their nuclear weapons exist to blackmail other states, they aren´t at risk of survival, they don´t have non-first use and they will use their nukes even in a convectional war. for example The U.S. and Russia.
2- Minimal Deterrence: They have non-first use, the maintain an arsenal to ensure minimal deterrence and they will only use their nukes if threatened with nukes. For example China and India.
3- Survival Arsenal: These are states will only use their arsenal if they perceive that their country survival is at risk. For example North Korea and Pakistan.
I think Japan Right Wing Parties are pushing these policies because probably they think "nukes are cool", I wouldn´t be surprise is that is the case, these parties are like magatards, and second because they probably THINK they could use them as weapons of coercion, without really understanding what that entails and the real danger that could bring to their own country and their own people. Japan is under US nuclear umbrella and have ZERO existential risks in Asia, what is the point.