The integration of reconfigurable intelligent surfaces (RIS) into multiple-input multiple-output (MIMO) systems holds transformative potential for enhancing spectral efficiency (SE). However, practical deployment faces critical challenges stemming from hardware impairments, particularly the nonlinearity of power amplifiers (PAs) at the MIMO transmitter, which causes signal distortion that propagates through the RIS-reflected channel. To bridge this gap, this paper investigates a holistic hardware-aware design that jointly optimizes the RIS phase shifts and the base station transmit precoder under a realistic nonlinear PA model. We propose a joint optimization framework that explicitly balances beamforming gain and nonlinear distortion suppression. Specifically, we develop an alternating optimization (AO) framework in which the fixed-precoder RIS phase design is formulated as a constrained unimodular fractional quadratic program and solved via a Dinkelbach-based projected gradient ascent procedure, while the fixed-RIS precoder design is refined through a successive convex approximation (SCA)-based iterative update. Furthermore, to address latency- and complexity-constrained scenarios, we develop a low-complexity RIS-aware zero third-order distortion (Z3RO) joint design approach, in which the BS precoder inherits the Z3RO distortion-nulling structure through the effective cascaded channel and is combined with iterative RIS phase adaptation. Simulation results demonstrate that the proposed methods can achieve at least a 36.4% increase in SE over conventional benchmarks under PA nonlinearity.