I'm trying to do a search.lookupFields to find an Item Category based on the Class ID, but I don't know what to use as the 'type' for the lookupFields. I've tried class, classlist, and classtype, but none of them are working. Any ideas?

I'm actually not entirely sure what you're trying to achieve, but you could check the search.Type enum in the manual for the full list of types

I was trying to get at the Class list under Setup -> Company. Turns out the type was search.Type.CLASSIFICATION