To determine what type of carrier it should be, I would consider what kind of job you expect it to do. Consider these US Navy WW2 era standards:
Fleet Carriers were the largest carriers that served as the flagships of entire carrier battlegroups or fleets. They carried a large number of craft allowing them to fill almost any role. Most of today's carriers fit this catagory.
Light Carriers were basically the same thing but much smaller, cheaper, and easier to produce. In the sudden naval buildup of WW2, it was easier for the US to field 3 of these carriers than one Fleet Carrier.
Strike Carriers were medium sized carriers designed to carry naval attack and bombing craft almost exclusively. They were assigned the task of destroying enemy shipping, important land targets, and supporting ground troops in the US' island hopping campaigns. They carried few dedicated fighter craft.
Attack Carriers were a smaller very fast type of carrier designed to get into and out of operational areas quickly and hopefully before enemy forces could respond. They acted very much like small, fast Strike Carriers.
Escort Carriers were numerous in US Naval campaigns and carried fightercraft exclusively. They were built on fast Attack Carrier hulls and their sole purpose was to defend other warships and convoys from air attack.
Consider these and you may find your answer.
As for a class name, what about Midway or Essex?