Just remember that the Alaskas managed to put 9x 12 in guns to sea on a displacement just over 30000 tons. I have to draw attention to the fact that the Scharnhorsts only managed to put 9x11 in guns to sea on 38000 tons. The armouring scheme of the Scharnhorsts was more extensive, but not greatly so, ....the turrets on the Alaskas for example had armour plate 11-13 inches thick, whilst the turerets in the Scharnhorsts were 14 inches on the front facings and only 6 inches elsewhere. . The main belt was 6-13 in the Scharnhorsts compared to 5-9 inches in the Alaskas
The Alaskas are some of the most misunderstood ships in History. They are often referred to as Battlecruisers in popular literature, but the USN never rated them as such. Conways describes them as the logical development of Heavy Cruisers, free of all treaty restrictions. They were never intended to fight in the US gunline, like Battlecuisewrs were designed to do
I certainly agree that the Scharnhorst weighed more than the Alaska but I don't think the difference is as great as those mentioned in your posting.
I have
Scharnhorst Standard 34,800 tons, full load 38,900 tons
Alaska Standard 29,800 tons Full Load 34,250 tons.
Against this the Scharnhorst had much better armour. As you mention the Scharnhorst had a much thicker main belt by a significant margin. The Main Turret Armour was very similar
Alaska 12.75 in Face, 5 to 6in on the other areas,
Scharnhorst 14 in face, 6 in to the others.
I am as sure as I can that the extra weight went into the armour and the secondary weapons. 12 x 5.9 and 14 x 4.1 will weigh a lot more than 20 x 5in.
As for the titles these are more or less meaningless. At the end of the day the Dunkerque was designed to face off against the Graff Spee as was the Alaska (source American Battleships, Carriers and Cruisers ISBN 35601511 4) and the Scharnhorst was designed to take on the Dunkerque, so they share a similar bloodline. That the USA decided to use an enhanced Cruiser design is a logical and effective approach to solve a problem and they ended up with a good solution.
Juha
No battlecruiser was supposed to stand in the battleline and slug it out against a Battleship. To do so is asking for trouble.