This paper reports the development of a novel bio-inspired composite sandwich panel (BCSP) with fiber metal laminate (FML) face sheets and a dual core to improve the low-velocity impact behavior based on the woodpecker’s head layout as a design template. The dynamic response of BCSP under impact load is simulated and analyzed by ABAQUS/Explicit software and compared with that of the composite sandwich panel (CSP) with a single foam core. The impact behavior of BCSP affected by these parameters, i.e., a different face sheet thickness, rubber core thickness and foam core height, was also reported. The results show that BCSP has superior impact resistance compared to CSP, with a lower damage area and smaller deformation, while carrying a higher impact load. Concurrently, BCSP is not highly restricted to any particular region when dealing with stress distributions. Compared to CSP, the bottom skin maximum stress value of BCSP is significantly reduced by 2.4–6.3 times at all considered impact energy levels. It is also found that the impact efficiency index of BCSP is 4.86 times higher than that of CSP under the same impact energy, indicating that the former can resist the impact load more effectively than the latter in terms of overall performance. Furthermore, the impact resistance of the BCSP improved with the increase in face sheet thickness and rubber core thickness. Additionally, the height of the foam core has a notable effect on the energy absorption, while it does not play a significant role in impact load. From an economic viewpoint, the height of the foam core retrofitted with 20 mm is reasonable. The results acquired from the current investigation can provide certain theoretical reference to the use of the bio-inspired composite sandwich panel in the engineering protection field.