This paper investigates the performance of a time-switching (TS) based cooperative non-orthogonal multiple access (NOMA) network consisting of a base station (BS), a near user (NU), a distant user (DU), and a full-duplex (FD) relay. The BS shares a direct link to the NU, while communication to the DU is assisted by a battery-aided energy harvesting FD relay. We consider a static battery energy (SBE) scheme in which a fixed amount of battery energy augments the harvested energy and analyze the performance of both NU and DU. Also, a new dynamic battery energy (DBE) scheme, wherein the harvested energy is augmented with as little battery energy as possible in order to achieve desired quality of service, is proposed and is shown to ensure efficient battery utilization. Considering a threshold-based nonlinear EH model, closed-form expressions are derived for the throughput of NU and DU for both SBE and DBE schemes. Furthermore, for the DBE scheme, we also derive an exact closed-form expression for the average battery energy drawn per symbol interval. We then demonstrate that the choice of battery energy and TS parameter is crucial to attain a maximum DU throughput while simultaneously guaranteeing a target throughput at the NU.