I like the method of using Use Cases or User Scripts to generate tests. How is this class to be used?
If the class is too big to test that way, maybe it should be a collection of sub-classes, or, as you have said yourself, maybe the design is at fault.
To be a real class, it has to have encapsulation, and therefore an interface. From there, the tests become simple: does the interface work as advertised?

This sounds like an interesting project.