A fighter carrier in this environment is just plain silly, though the idea of parasite craft docked to the outside of the ship Suliban-style would work extremely well.
Is it not still a carrier?
Besides, a shuttle bay works pretty well, why coudn't a ship house fighters and launch from that platform?
Not sure why it's silly
The ultimate fights begins. Silly would be the ideal of massed squadrons of piloted craft shooting out of Enterprise. Since they would be unable to rundown D7 cruiser because fighters in the Trek verse are not fast enough and don't have enough endrance to chase a starship down. Its not Battlestar where the only option is to jump to hide, Enterprise can outrun the shuttle with a phaser and a torpedo or two which we want to call fighters
The ideal of a parasite is like the float planes of old earth navies. A scout ship, not the decisive weapon which airpower holds because of speed advantages on earth over waterborne navy ships.
I wasn't speaking of the Enterprise exactly... here's my initial post on this subject...
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Seeing as the first warp crafts were essentially planes (besides the "Phoenix" and I assume similar craft to test warp drive), I can imagine that early designs were fighter type crafts with jet engines and small warp nacelles. In deep space, a first line of defense squadron might also have been needed in some missions to protect the cruiser before photons, sensors, and shields were fully developed. Maybe just a handful of fighters in the hanger bay just in case, or as scouts flying ahead of the main ship, or planet recon. Although Trek history has changed, details of the first Romulan war depicted the use of nuclear weapons. Archers Enterprise just had hull plating... so I think fighters could have come into play. I could see starfeet "space craft carriers" with small fighters and other weapons platforms engaged against a Romulan threat.
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Take into account that I used the world "imagine"... I'm engaging in playful speculation. But anyway, I was talking about early designs of warp capable craft, and how in the those days (before the fruition many of the trek advancements already discussed) it may have been interesting to think of the usefullness of a fighter. Before large cruisers, and as Earth based warp fighters and such...
But beyond that, the idea of a warp capable fighter could be cool... perhaps a few even stationed at a Federation planetary base, or even a small star base for protection when no other ship can respond in time to any given threat. Not just escorting larger ships or carriers or as part of an armada. If they do run into trouble, instead of ejection seats they could have emergency transporters, or "beam out" levers. Think of the action scenes of 3 or 4 Starfleet fighters taking on an Klingon bird of prey attacking a Federation colony... A Trek style dog fight !!!