Imo the Heron is the best pure carrier right now: 3 bays, a good fighter boost system, and high enough speed to kite cruisers and slow destroyers. But like Supraluminal says because of the AI tags its not suitable for a battle carrier build outside of player hands. The AI simply won't fly it right, which is a pity and not told to players in any way.
In player hands it has a comfortable amount of OP to be flown with a heavy blaster and mid-cost fighters combining kinetic and EMP. Its not a slugging battlecarrier, but it can be used to help more quickly finish off targets that its fighter has swarmed or to overpower weak destroyers by itself. The main key is to not install expanded flight decks (expanded flight decks is overrated and overcost and not needed for many fighter compositions, though it helps for bombers). DTC/ITU is also not required or even good if not being built in. This might seem bizarre, but for something as fragile as the Heron who's primary armament is its fighters range isn't important and with 80 base speed the ship quickly closes over the distance that the DTC/ITU bonus covers on a 600 range weapon. The ship is moving in to help support the fighters and provide armor breaking/dps after they've fluxed up the target and hopefully EMPd it as well, not engaging in a gun duel. Its fine to build in, but not especially needed.
The biggest thing that helps carriers is taking the carrier skills and CR boosters: +75% faster fighter refit time (for an officered ship) alongside -10% refit time from CR is very good for keeping the carrier's rate up; -10% damage received, +10% damage to all targets, perfect leading accuracy, and +40% speed on the fighter end make them much more effective and survivable.
[Edit] Decided to do a few mission screen (so no skills) builds to see what works well. Here are a few:
3x Thunder, 1xHeavy Needler, 1x antimatter blaster, 1x Ion Cannon, 2x pd laser (rear), 20 caps, 12 vents. No hullmods. Engage fighters at long range with system and close: once target distracted get into knife fighting range and pull back to regroup to preserve replenishment. Can easily take on multiple destroyers or a cruiser though it takes a while to chew through hull. The Ion Cannon can be swapped for a 2cd am blaster, but flux starts to get a bit tight so thats probably for campaign where skills boost it. Changing fighters to Broadsword-Broadsword-Claw makes them less good vs frigates but better vs larger and saves some OP for flux. Both changes together is 2xBroadsword, 1xClaw, 1xHN, 2xAMB, 2xpdl, 20 caps, 15 vents. Gladii can be substituted for broadswords: they are more fragile, but cheaper, faster (claws are faster too) so better vs small ships/more responsive to commands, and have higher firepower than broadswords.
Classic heavy blaster works and cuts through hull fast, and is OP cheaper, but lacks the shieldbreaking of the heavy needler. This works: broad/broad/claw + HB + burst pd laser (front) + 2x pd laser (rear) + 19 caps + 30 vents. Gladii instead of broadswords is the same consideration as before - better vs small craft including destroyers, worse vs large. The flux cost imo makes this build not very good without skills, but with them it works better.
The nemesis of the battlecarrier Heron are Hammerheads (in numbers, a lone one is dead pretty fast). They are just fast enough to kite (without speed boost skills/officers), have high firepower so shoot down fighters, and have high enough armor/hull that it takes a while for the 3x wings to overwhelm them individually. Other fast destroyers can be troublesome but are much weaker vs fighters so don't pose as much of a threat.