Massive multiple-input multiple-output (MIMO) technology has attracted much attention in low earth orbit (LEO) downlink satellite communication (SATCOM) systems recently since the energy efficiency (EE) and spectral efficiency can be significantly improved. In order to reduce the power consumption for the massive MIMO LEO SATCOM systems, we focus on the hybrid analog/digital architecture in this work. Considering the limited resolution of the phase shifters in practical MIMO SATCOM systems, a twin-resolution phase shifting (TRPS) network is proposed to make a meticulous tradeoff between the power consumption and array gains. In addition, we examine the impact of the distortion, introduced by the power amplifiers (PAs) to the system design, by considering nonlinear PA models. Moreover, we propose an efficient algorithm for TRPS-based hybrid precoding with nonlinear PAs. Numerical results show the EE gains considering nonlinear distortion and the performance superiority of the proposed hybrid architecture compared with the baselines.