Hybrid energy supply composed of batteries and radio frequency (RF) signals has been anticipated to be a prominent solution for balancing the reliability and self-sustainability of future IoT networks. The newly emerging reconfigurable intelligent surface (RIS) is also capable of greatly enhancing spectral and energy efficiencies. In this paper, by considering an asynchronous transmission protocol among all energy receivers (ERs) and assuming the perfect self-interference cancellation (SIC) at the hybrid access point (HAP), we aim to maximize sum throughput in the RIS-aided hybrid powered communication networks (HPCNs) by jointly optimizing the transmit covariance matrices of the HAP and all ERs, the RIS reflection matrix and the downlink/uplink (DL/UL) time allocation. Generally, this optimization problem is intractable to solve due to strongly coupled variables and nonconvex unit-modulus constraints. To draw more insights into this joint design, we firstly carry out feasibility analysis on this problem, and then develop a 2-block alternating optimization algorithm, which consists of the semi-closed-form solution based iterative algorithm for deriving the optimal MIMO transceivers together with the DL/UL time allocation and the alternating direction method of multipliers (ADMM) based algorithm for the RIS design. To avoid the potential high complexity of alternating optimization, we also propose a two-stage scheme, where the RIS design is independent of the others and aims to create favorable DL/UL channels. The extension of our proposed algorithms to the practical imperfect SIC case is then discussed. Numerical results illustrate the superior performance of our proposed algorithms over the baselines in terms of the achievable sum throughput, and their time effectiveness in solving large-scale problems.