Thanks for your explanation of the rationale behind the RNs choice of 12" guns for the Invincibles. And noting that while they were a flawed concept it wasn't just inadequate Armour that caused the losses at Jutland but poor practice. Something independent of the design flaws.In 1905 the US was building armoured cruisers with 4x10 inch guns, the Japanese were building them with 4x12 inch guns, both classes about the same size as pre-dreadnought battleships. Given this a fast all heavy gun ship with 10-12inch guns about the size of the dreadnought was a natural step, and after comparing 10inch and 12inch guns the British determined 12inch was the better choice (similar rates of fire, dramatically greater range & hitting power). It's possible that battlecruisers were nearly as inevitable as the dreadnought and it's just that the British got there first (before the Japanese?). In addition there was an operational reason for the British to build them, they (or some of them) believed that wireless telegraphy (still very new) and their ability to tap international communications cables in wartime meant that they could protect their commerce with small numbers of very fast and powerful ships that could move to wherever needed. At the same time the new fire control procedures being developed were thought to be going to enable the British to hit the enemy at ranges at which they wouldn't be able to respond effectively, making armour unnecessary as long as the British held a speed advantage. Smaller scout cruisers would also be needed to scout for the battle line, but would in turn need protection from larger cruisers if they were to maintain contact with the enemy while the main fleet closed.
While the idea of such large and expensive ships that were so vulnerable against their equivalents can be seen as fundamentally flawed, it's worth noting that in practice the British losses were mostly due to bad shell handling procedures and cordite rather than the design of the ships themselves, and that even so they carried out the role assigned to them pretty well.
Whether the Germans needed battlecruisers is an interesting question, they were too large to be effective open ocean commerce raiders (coal supplies and geography) and I think the Germans had less need to track the British fleet if all they intend is to evade. On the other hand, a squadron of fast heavy ships was very useful in the North Sea as their speed cut down the time available for the British to react, and might also have been useful if the British had imposed a close blockade.
The snag with giving battlecruisers the same caliber armament as your battleships is that they become capital ships in everyone's eyes. Not just to be used as a fast wing for the battle line but ships whose loss is a sharp blow to prestige and the navy's strength. Whereas cruisers are essentially expendable. So it might have been better to stick to the 9.2" caliber of the armoured cruiser breed for the single caliber design. The cost savings might not be much (though every little helps when you're dealing with the U.K. Treasury) but maintaining the distinction between the battle line and your cruiser force would be important. You could raise the caliber to 10" guns for the follow on design contemporary with the Orions and KGVS, and even 12" for the Renowns. Or simply build more 15" gun fast battleships like the Queen Elizabeths.
Anyone know how Invincibles with 9.2" guns would have come out on weight and cost?