Why a late-war battlecruiser still needs battleship guns

Should an ultimate late- or postwar battlecruiser carry 12-inch or 14-inch guns?

No; an Allied design would need at least 15-inch and probably 16-inch guns, plus enough speed to distinguish it from fast battleships.

A 12- or 14-inch Allied battlecruiser would be too lightly armed for the period. Scharnhorst carried 11-inch guns because German industry could not yet supply the intended 15-inch replacement, while Alaska used 12-inch guns because she was a cruiser. A British late-war battlecruiser would resemble a longer, faster Vanguard with 15-inch weapons, and an American one a stretched Iowa with 16-inch guns. Fast battleships already made 28-30 knots, so the battlecruiser would need still higher speed and firepower credible against the capital ships it could not avoid. The historical relationship between battleships and battlecruisers makes equal-generation main armament the normal expectation.