Polymer-based electrolytes potentially enable enhanced safety and increased energy density of lithium-metal batteries employing high capacity, transition metal oxide-positive electrodes. Herein, we report the investigation of lithium-metal battery cells comprising Li[Ni$_{0.6}$Mn$_{0.2}$Co$_{0.2}$]O$_{2}$ as active material for the positive electrode and a poly(arylene ether sulfone)-based single-ion conductor as the electrolyte incorporating ethylene carbonate (EC) as selectively coordinating molecular transporter. The resulting lithium-metal battery cells provide very stable cycling for more than 300 cycles accompanied by excellent average Coulombic efficiency (99.95%) at an anodic cutoff potential of 4.2 V. To further increase the achievable energy density, the stepwise increase to 4.3 V and 4.4 V is herein investigated, highlighting that the polymer electrolyte offers comparable cycling stability, at least, as common liquid organic electrolytes. Moreover, the impact of temperature and the EC content on the rate capability is evaluated, showing that the cells with a higher EC content offer a capacity retention at 2C rate equal to 61% of the capacity recorded at 0.05 C at 60 degrees C.