Can battleship guns make a carrier self-defending?

Is a battlecarrier a sound way to let an aircraft carrier defend itself against enemy battleships?

No. If an enemy battleship reaches gun range of a carrier, the screen and the carrier's operating concept have already failed in more important ways.

Heavy guns do not solve a carrier's central defensive problem. Carriers derive value from launching aircraft at distance while escorts, scouting, and command keep enemy surface forces away. Historical cases in which battleships caught carriers at gun range occurred after multiple prior failures; adding a battleship battery would consume space and resources without repairing those failures. Better command accountability is a cheaper remedy for reckless carrier handling than building every carrier as a capital-gun hybrid.