The bistable piezoelectric composite laminates have two stable states, which can undergo the nonlinear large amplitude vibrations, namely the snap-through vibrations, are used to the broadband energy harvesting. This paper proposes a bistable energy harvester for the bistable piezoelectric composite laminates with the simply supporting of four corners. The nonlinear vibrations and energy harvesting are studied for the bistable piezoelectric composite laminates. It is determined that the piezoelectric layer attached to the asymmetric bistable four corner simply-support (FCSS) composite laminate still maintains two stable-states. The energy curves of the system have two potential wells, but it is not symmetric. The displacement bifurcation diagram, voltage bifurcation diagram and corresponding maximum Lyapunov exponent are simulated. The waveform, phase portrait and Poincaré map are utilized to illustrate the complex nonlinear dynamic phenomena of the bistable FCSS piezoelectric energy harvesters. The forward sweeps and backward sweeps are performed to identify all-around dynamic behaviors. The output voltage of the bistable piezoelectric FCSS energy harvester is evaluated. Analyzing the amplitude frequency response curves for the displacement and output voltage under different excitation amplitudes, the optimal frequency range of the bistable piezoelectric FCSS energy harvester is determined. As the excitation amplitude increases, the frequency bandwidth of the snap-through vibration also increases. Due to the soft spring characteristics and hysteresis effect of the system, a significant deviation is observed between the forward and backward frequency scans, indicating that during the backward frequency scan, the limit cycle vibration expands the frequency range of the double-well vibration. The vibration testing platform for piezoelectric testing is built and the voltage signal of the bistable piezoelectric FCSS composite laminate is tested.