Really? Japan developing Nukes in wartime and launching towards China?Because that "Iran style" worked so well for Iran, right? No wait, they were bombed to kingdom come, hmm...
If you're an island nation and don't have a potent navy, with surface combatants, support vessels and submarines, you're not seriously trying. Carriers, especially these smaller ones, have much more utility than ballistic missiles. Be it disaster relief (very important in the region) as well as supporting amphibious operations in the region against non-peer factions, conducting reconnaissance as well as being a platform to launch ASW and AEW&C style aircraft. All of these points are valid reasons to invest into medium sized flat decks. I think you can't quite grasp the utility of a ship that can launch and host several large helicopters. The ASEVs are ABM ships, so while very specialized in essence, their role is very important given the potential means of a Chinese attack. And if one is like Japan and has genuinely decent shipbuilding, then it's a no brainer to invest into powerful and versatile naval vessels.
If the PLARF would attack mainland Japan or attack the JMSDF in a large scale, they'd catch nukes anyway so it's a moot point (as these mass attack fantasies are always) because these would either come from the US or Japan directly, which as a nuclear latent state, could more or less develop the bomb in weeks, if not days.
And why would China allow that to happen? If China gets any indication that Japan intends to do that, then the most logical next step will be to Nuke Japan first before it can threaten China.
If you are not already a nuclear weapon state, then don't threaten to develop a nuke to Nuclear superpower, cause they can also nuke you first.