From my brief time using PyStan some months back, pickling the fit object seemed to work. However, when I did it, there was a warning message indicating that this was an experimental feature of PyStan, and that to one should unpickle the model associated with the pickled fit object before unpickling the fit object itself.
What I ultimately ended up doing (before switching to RStan) was to pickle certain outputs from the fit object, such as the dictionaries returned from running fit.extract() and fit.get_sampler_params().