This study proposes a novel topology optimization framework designed to handle unsteady fluid–structure interaction (FSI) problems involving large deformations. FSI phenomena are ubiquitous in engineering applications, ranging from aerospace components to biomedical devices. However, designing structures that effectively exploit FSI remains challenging due to complex transient behavior. Conventional FSI topology optimization methods compute the solid domain based on Lagrangian or ALE approaches, which are primarily limited to small strain problems assuming linear elasticity due to mesh distortion issues. In contrast, the proposed method employs a fully Eulerian approach, which fundamentally prevents mesh distortion. This approach ensures high robustness and computational efficiency in topology optimization by uniformly describing fluid and solid domains on a fixed Cartesian grid. A key feature of this study is the regularization of the left Cauchy–Green deformation tensor, which guarantees the differentiability of the governing equations at the fluid–solid interface. This enables the derivation of consistent sensitivity analysis using the continuous adjoint method. The optimization problem is formulated as an inverse problem to determine the optimal initial material distribution that maximizes performance during time evolution. In the presentation, several numerical examples will be provided to demonstrate the effectiveness of the method in optimizing FSI systems under dynamic fluid forces.