I am curious about the AI strategy flag and how that affects bombarding. Specifically, that the CCM carrier settings may actually be ideal for battleships.
In CCM, the carriers as you described are there to allow the player to have the traditional civ-style carrier as well as a bombarding role. The AI strategy is important in how the computer uses it. CCM bypasses the issue of how the AI strategy affects build priorities since the carriers are auto-produced. But I think the Naval Carrier strategy alone would be good for both carriers and battleships. The computer will bring three escorts along with units that have the carrier or transport strategy, even if they don't have the "Requires Escort" flag. I have seen this behavior in several mods. If the ship is flagged as Naval Power, it runs the risk of being turned into a meager escort for some transport. For AI battleships, not only does that squander their bombard, but it also doesn't even protect the transport since they are king units.
As Naval Carrier-flagged units, they would have their own escorts. And if the AI is indeed using bombardment on ships with the Naval Carrier flag (albeit unreliably), then this setting may prove better for both battleships and carriers. Of course, this doesn't help with bombard ranges over 2 as we know the computer is not programmed to reconnoiter its targets. But perhaps the AI could more consistently bombard with capital ships (possibly to include battlecruisers and heavy cruisers) if the Naval Power strategy is left to the destroyers and light cruisers.
For example, in the above photo you provided, two battleships and a carrier are escorting a transport. Did they sit there like gormless lumps or did they bombard the city? I know that was a long time ago, but it seems to me that ships used as escorts by the AI fail to use their bombardment. In WW2 games I often see a stack of say 6 enemy ships, where three are escorts, 1 is a transport, and two provide shore bombardment. A human player has an advantage as he will always use his big ships for bombarding as well as escorting valuable transports.