State estimation for nonlinear dynamical systems remains a fundamental challenge, particularly when measurements are sparse and internal states are inaccessible. This work presents a KOOPMAN-based Linear State Observer (KOOPMAN-LSO) design framework that enables linear observer synthesis for nonlinear systems through KOOPMAN operator theory. The nonlinear dynamics are lifted into a higher-dimensional observable space using physics-informed basis functions, where a linear predictor with control is identified via extended dynamic mode decomposition with control (eDMDc). A discrete-time Luenberger observer is then constructed in the lifted space, and the observer gain is obtained through a dual linear - quadratic regulator (LQR) formulation to ensure stable and tunable estimation error dynamics. The proposed framework combines the representational capability of KOOPMAN lifting with the simplicity and computational efficiency of linear observer design, providing a systematic approach for nonlinear state estimation under limited sensing. Its effectiveness is demonstrated on a latent thermal energy storage (LTES) system based on phase-change materials (PCM), where internal temperature states are not directly measurable. Experimental results under varying operating conditions show accurate reconstruction of unmeasured states from limited output measurements, illustrating the potential of KOOPMAN-LSO design for practical nonlinear systems. The proposed approach achieves high-fidelity reconstruction with an RMSE as low as 0.0819 °C for the LTES outlet temperature and generally below 1.0 °C for observable internal PCM temperatures.