Changing the source-detector geometry will not affect the detector resolution. (However, it can increase the number of counts in the peak and thus improve the accuracy of the estimate of the resolution.)
You cannot do proper pulse shape discrimination (PSD) by measuring energy deposition, because the shape variation is due to the difference in time evolution of scintillation processes between neutrons and gammas. You need to use the optical package so that scintillation and optical photon transport processes are involved. Then you can save the time when each optical photon hits the face of your scintillator volume adjacent to the photocathode end of PMT in SteppingAction. The time distribution can be saved in a Root Tree file or .csv file. Those histories are best processed off line in a separate program like Octave or Root to do the actual PSD.
This results in PSD of the pure scintillator response. If you want to include the response of the PMT, I would recommend doing that in Octave or Root as well, since Geant4 is not really suited to accurately building the mechanical and electrical aspects of a PMT from first principles.
You could also use the G4VDigitizerModule class to simulate ADCs, TDCs, etc. in your PSD scheme. I have never used it, so I cannot comment on its usefulness. (See Digitization - Book for Application Developers)