The hybrid photovoltaic (PV) with energy storage system (ESS) has become a highly preferred solution to replace traditional fossil-fuel sources, support weak grids, and mitigate the effects of fluctuated PV power. The control of hybrid PV-power systems as generation-storage and their injected active/reactive power for the grid side present critical challenges in optimizing their performance. Therefore, this paper introduces hybrid PV-battery parallel inverters employing a finite control set model predictive control (FCSMPC) method. The proposed FCSMPC-based controller and inverter system achieves multiple functionalities, including maximum power extraction from PV, proper charging/discharging commands for ESS, support for weak grid conditions, support during low-voltage ride-through (LVRT) by increasing reactive power injection to counteract the drop in grid voltage, and economic management based on feed-in-tariff (FiT). The controller significantly improves the performance of the PV-battery system under faulty LVRT conditions and unbalanced grid voltages, satisfying grid code requirements while continuously supplying the microgrid’s delicate local load. A real-time simulation hardware-in-the-loop (HiL) setup, utilizing the OPAL-RT platform, is employed to implement the proposed hybrid PV–ESS with its controller. The results affirm the superior ability of FCSMPC in weak-grid conditions and its capability to achieve multiple objectives simultaneously.