I question the usefulness of a proper carrier. The M.O. of our opponents has to be to sneak within medium range, then open fire with fusion guns. Carriers are good at defending against missile/torpedo attacks, and similarly, can launch missile/torpedo salvoes at longer range than a conventional ship. However, they are not efficient torpedo platforms close-in (since conventional platforms can launch torpedoes without the overhead of carriers' handling decks or extra crew - the new carrier hits about as hard as a CFA IMO). Patrol carriers were justified in that they can carry an EWAR squadron, but a bigger ship won't improve on that (and I'd argue in the future for a smaller "aviation cruiser" that can do the same while having a gun armament).
IMO, if we're going to lay down a capital ship of a new type, we should pick something that is meant to turn the tables on how the Lydians fight. We've had the greatest success when we let the Lydians get close, are designed to withstand the initial salvoes from their fusion bays, and then we open fire with our mass driver cannons and torpedoes on the now-spotted enemy ships.
What I'd suggest doing is designing a ship fitted for but not with our new, fancy weapons. To start with, the big upgrades for us at TL 10 will be turreted LMDCs on ships of >5000 tons, turreted HMDCs on ships of >10000 tons, and point defence batteries. The LMDCs are an easy swap, I think; we can put the turrets on, but then fit them with a missile bay instead of the mass drivers. Once we find a supplier for non-prototype LMDCs, we can replace the missile bays. Yes, we're wasting 5 tons per bay, and a additional 50 tons over the entire ship (for a 5,000 ton ship), but that's really not that much when it gives us growth potential. Alternatively, we could build a 10,000 ton ship and use LMDCs for now and HMDCs later. Point defence is a bit harder, but we can do that by designing the ship with the crew and weapons we'd use if we had PD batteries, then removing them and adding virtual crew-controlled PD turrets until we hit the cap; spare hardpoints post-upgrade can be fitted with the zero tonnage missile racks carrying point defence missiles.
That said are you sure we have 10,000 tons free? I thought we had less...