Which is where ASBM becomes a very interesting idea. Not particularly expensive to buy, very cheap to operate, extremely hard to destroy on the ground (the range vs. theater size gives you a lot of China to hide in). Developing the RV is the biggest job. The technology is only a step or so beyond Pershing II, and you can dial-in a lot of different effects, such as a load of deck-puncturers that would leave the CVN afloat but with the cats and elevators hosed.
The hard bit is the targeting, although these days even small UAS can do that, not to mention satellites like Kondor-E. (There was a model of a truck-mobile satellite launcher at Zhuhai in 2014.) Of course, your targeting isn't really credible until you've got a realistic target to practice on. Which China now has.
I've got a few caveats.
Drones: It would seem that using drones to do the targetting for a ballistic missile would not work. Drones are relatively easy to shoot down and being unmanned there would be no hesitation to shoot one down if the political/military climate between the shooter and the target nations got warm, never mind hot.
Satellite tracking: Yes, satellites can track a carrier. But carriers are mobile and satellites have only relatively small windows of opportunity to track any specific area. These windows are predictable and are spaced relatively widely apart. It would take a truly massive constellation of satellites to generate data sufficiently precise, accurate, and timely enough for weapon targeting of a mobile target like a carrier.
Deck Puncturers: A ballistic missile reentry vehicle with a terminal guidance system accurate and precise enough to track a mobile target and a maneuvering system able to hit such a target? Has anyone anywhere ever made one of those? The Pershing II that was referenced had active radar homing and a CEP of 30 meters against a fixed land target. And that radar system used scene correlation, which would be useless over the ocean and useless against a mobile target. And assuming such an RV was even possible, could a ship's close in defense system deal with it? And assuming such an attack could be done successfully, would the reaction/consequences to a deck punctured carrier be much less than a sunk carrier? If not, why bother?
Missile defense: The Aegis system with SM3 and SM2 block4 is quite competent at shooting down ballistic missiles. Both Japan and USA have them in that part of the world and Taiwan is reportedly getting some also. And it would seem that in the event a ballistic missile was fired at a carrier the assumption is going to be that it carries a nuke. And if such an attack failed it is going to result in an awful lot of hurt for the country that fired that missile.