This paper considers a nonlinear suboptimal control problem for a near-space hypersonic vehicle's (NSHV's) attitude dynamics. The least-square and stable manifold methods first solve an unconstrained approximately optimal control law corresponding to the nonlinear attitude model. Then, to further meet the dynamic performance requirement of the attitude control system, a novel strategy based on the Koopman operator, symplectic geometric theory, and the stable manifold theorem is proposed to approximate the eigenvalues of the closed-loop nonlinear unconstrained approximated optimal control system. The weight matrices in the optimal performance index, which directly determine the output responses of the nonlinear attitude dynamics, can be appropriately designed according to the eigenvalues. The final control law considers the actuator constraints. The NSHV's closed-loop attitude control system is proved to be locally exponentially stable, and the suboptimality of the control law is analyzed. Numerical simulation demonstrates the effectiveness of the proposed scheme.