Can the sea sparrow and/or RAM even launch in time to shoot down the shells? Looking at a youtube video launch, it takes a about second to even launch one of the darn things. I'm pretty sure in real life situations the missiles are designed to shoot down projectiles launching from many miles away, and would therefore have enough time to track and engage them. From 1 mile (~1600 meters) away, the shell traveling at roughly 750 m/s would have a flight time of something between 2.2-2.7 secs. It doesn't even matter if the battleship has to aim first, because you can't fire the missiles to intercept the shells until the battleship actually fires. Based on that, I would say the point defense of the carrier is useless in this situation.
Also note that the carrier itself carries no anti-ship armament. I suppose you could launch the sea sparrow at the battleship (assuming the missiles, which are designed to destroy fighters, could even damage the battleships), but the shells would still arrive first, because the missiles have a much longer acceleration time.
Another point we can address is whether or not the CIWCS system would be sufficient to stop the shells. Frankly, I have no idea whether or not a stream of bullets can prematurely detonate a 1200kg shell, so I can't say. I would bet, however, that it can't, since it's really only designed to shoot down relatively flimsy missiles.
Also there's the question of how accurate the battleship could be. Again, here I can only guess. The guns are aimed by radar, and have computers to compensate for the carrier's speed and range. Since the travel time of the shells is about 2.5 sec, there really isn't enough time for the carrier to alter course in order to dodge fired shells, so I feel like that isn't an issue. I would say, that since the guns have a max range of 24 miles, a 1 mile target should be relatively trivial. Also keep in mind the battleship can fire twice a minute, so worst case scenario is that the battleship puts out 2 salvos before the carrier's aircraft can fire anything at it.
Humorously enough, the battleship does actually have antiship missiles in it's armament, so I guess that's also a factor. With the point defense of the carrier to take into consideration, I can't say what the result would be.
Finally, if the battleship did manage to hit the carrier with even one shell, I would say the battle is over at that point. Carriers aren't armored to survive 16 inch shell fire, because it's not supposed to stand up to that kind of firepower in the first place.
Given all that, I believe the battleships *should* be able to sink the carrier, although whether or not the carrier can launch at least one plane to try and sink the battleship in return is beyond what I can say.