However, using other included libraries does not pose a problem. The org.json library is fairly small and standard, so I’m curious why this one is causing this issue?
The problem is that json-20141113.jar is included as a single file into your jar. The jar you deploy into API Engine must be flattened, that is all dependencies must be expanded within the jar (aka fat jar).
I also can suggest you building your projects with Maven Shade plugin. It creates one jar with all unpacked dependencies.
Please, take a look at my example.