Generally speaking, WWII carriers fell into four types: fleet carriers, light fleet carriers, escort carriers and seaplane carriers. The term “heavy carrier” wasn’t really used, as I recall. The differences between the four types were basically as follows. Fleet carriers were fast, and they operated the full range of carrier plane types; both these characteristics also applied to light fleet carriers, but they were smaller and thus they carried a smaller numbers of planes than full-scale fleet carriers. Both types were suited for offensive action against enemy fleets. Escort carriers were slower, and they operated a more restricted range of plane types; they had primarily a support function rather than an attack function. Seaplane carriers were the least capable of all; I’m not even sure that deck landings could be performed on them.
As I recall, Japan used a total of seven full-scale fleet carriers in combat during the war: the “original six” which attacked Pearl Harbor (Kaga and Akagi, Hiryu and Soryu, and Zuikaku and Shokaku), plus the heavily-armoured Taiho which became operational later in the war. I’m excluding Shinano from this list because, as Pacific War mentioned, she never reached a fully operational status and was sunk without seeing combat. So in my opinion, those are the seven carriers from which HBG ought to be choosing when deciding which one to produce as a first-rate carrier unit for Japan. I’d be fine with any of them as HBG’s selection.