I was playing the "what could you do with the Yamato class budget game and went with what would have been most useful given the war as we know it. Here's my idea: a ship of between 15,000 and 20,000 tons with 3 twin 12" turrets in the Tone arrangement. It would have the speed of the Kongos, torpedo tubes low in the hull, and decent armor against 12" guns.
This would give you the ideal ship for carrier escort (AAA and float planes on the aft part of the ships) and for fighting in the Solomons. I figure you could have had 4 by early 1942 and 6 by 1943. By simply replacing the Kongos they would have saved a fair amount of fuel oil.