What about like every Japanese/US CA of WW2, carrying array of 5in DP batteries along with their 8in main calibre (which is pretty much what you do in game)?
5-inch DP batteries are:
1.) Not a 6-inch gun, which is considerably more effective (DP secondaries somehow combine the firepower and piercing of a 6-inch gun with the AA of a 4 or 5-inch gun, which is a significant balance problem given that they sacrifice nothing).
2.) Lack the range of cruiser-caliber guns, meaning you can only use them at closer ranges (aka within effective torpedo range).
3.) Most batteries are around 2-8 guns per side (a lot less than I honestly expected). That's around a standard destroyer's firepower, which would imply that the cost-equivalent of destroyers would be significantly
better-armed than your heavy cruiser in light attack.
For comparison, if a 4-5 gun Mahan-class (they lost a gun in most refits) has an attack of 1.5 and a piercing of 2, they're going to bleed off most of their firepower against any armor, and require 3 guns to match the attack (and not piercing) of a single DP secondary. Even using DP primaries, 3 DP secondaries on a CA (and no CL guns) can do 13.5 damage before the 40% buff, for 18.9 damage and 8 piercing (making them effective against cruiser armor), versus 6 DDs with a DP primary (~1200 IC each means they cost more than a heavy cruiser before screens are added) only do 18 damage. 5 DDs with minimum guns screening that lighter-than-normal CA give it a further light attack of 5, for about 24 (equivalent to 8 DP primary-armed DDs, for ~9000 IC compared to ~9600 IC).
Compared to reality, where the heaviest secondary batteries on any ship were those of Yamato (per side, 9x6-inch and 6x5-inch guns, equal to 2 CL and 2 DD batteries, should give a total firepower of around 16 light attack assuming 1936 CL guns and DP light guns), the attack that a heavy cruiser can mount with just DP guns compared to triple 6-inch turrets (including 2 on the centerline) is somewhat insane, and most capital ships couldn't fit the centerline secondaries along with a proper main battery (conventional battleships in the US and UK had all their DP guns on the sides, for usually 10 guns per side or ~3-ish DP light guns for 9 light attack per side).
When you then see an all-6-inch heavy cruiser with 1 CA gun, 5 CL guns, and DP secondaries, you're talking about the equivalent of 4x5-inch guns, 24x6-inch guns (assuming ~5 from the 4-6 factor for guns-per-module), and 4x8-inch guns, you have the most heavily-armed cruiser ever conceived in the history of the world. This thing would need DOUBLE the length of a Brooklyn class, so you're probably looking at a cruiser at more-than 300 meters in length (versus 180m for a Brooklyn class, the most heavily-armed CL of WWII with 15x6-inch guns). For perspective, the Yamato measured 256 meters... it would require a battlecruiser hull to even fit the number of guns people spam on their cruisers, because some of the modules you put them on they wouldn't fit into (i.e. the extreme aft of the hull, or inside the main superstructure, given that most centerline turrets had to push other things out of the way).