We describe the experimental production of a beam of 23S1 positronium (Ps) atoms obtained from charge-exchange collisions between a positron beam and Xe held in a gas cell. The angular divergence of the emitted Ps beam was recorded using two position sensitive detectors located at different distances from the gas cell. The fraction of the Ps beam produced in the 23S1 level was measured via the change in the Ps count rate after driving the 23S1 → 23P2 transition with microwave radiation; with optimal experimental parameters, we estimate that up to 10% of the Ps beam is formed in the 23S1 state. The measured properties of the beam were used to evaluate the feasibility of using the system for precision spectroscopy of the n = 2 Ps fine structure using Ramsey interferometry