I can send the CAP to run him down
CVNs operate as part of a battle group. Said battle group includes anti-air assets, like Aegis, which can detect the drone and take it out. No CAP required. And with Hawkeyes in the air, the drone can be detected and killed long before the drone can detect the carrier.
The radar horizon of a radar at 50,000 ft is just under 275 NM.
Assuming the Hawkeye is at 30,000 ft and the drone is at 50,000 ft the Hawkeye's radar horizon to the drone is just under 450 NM. And that's assuming the Hawkeye is directly over the carrier. The Hawkeye will likely be stationed away from the carrier along the major threat axis. If the threat is China, that'd be between China and the carrier. And using cooperative engagement, the Hawkeye can guide a missile fired from an Aegis ship to a drone that's well beyond the ship's radar horizon. No CAP required.
So the battle group has the ability to take out the search drone long before it could track the carrier, and should that fail, to take out the missile in the exoatmosphere, and if that fails to take out the RV at high and medium altitude inside the atmosphere, and should that fail, assuming it's a deck puncher, to take it out in the terminal phase using close-in defense systems. So it's a very well layered defense.
And while it might take a finite number of sats to continuously cover the S. China Sea, it would still take quite a lot. And that assumes China would really focus that heavily on just the S. China Sea and not provide more global coverage.
And "complete lunatic?!!" Do you really think that if a nation punched a bunch of holes in a CVN that the reaction would be much less than if they had sunk it? Both are clearly acts of war, and once they've declared war in such a provocative way, all bets are off. And no matter how you look at it, a nation that declared war in such a fashion would be in a world of hurt. That does not mean a nuclear response (if that is what you were thinking), but their Navy is likely toast and their commercial sea commerce would be severely restricted if not outright cut off. And the base that launched the missile? Probably toast also.
And about your claim that USN believes China or somebody else has a ballistic missile with a deck punching warhead, care to share the source of that?