Integrating the attitude control and power storage subsystems of modern small satellites is of great significance in reducing the total cost of development and deployment in orbit. A combined energy and attitude control system adopts composite flywheels as actuators and energy storage simultaneously. Such small satellites should possess both stronger robustness and storage capability in the mission, especially confronting spatial disturbances. This article integrates sliding-mode control with fuzzy logic as a strategy to realize highly robust triaxial attitude stabilization under simultaneous solar radiation, gravity gradient, and aerodynamic disturbances while facilitating the momentum exchange of flywheels to improve energy storage capability. Unlike conventional nonlinear control, fuzzy logic can generate fuzzy rules to adaptively regulate sliding-mode switching gains. It strengthens system robustness in evidently decreasing angular velocities and control moments with finite-time convergence. Simulation results demonstrate the superiority of the proposed fuzzy sliding-mode control to a terminal sliding-mode control algorithm in combined attitude stabilization and energy storage, and also reveal the positive correlation of the control with energy storage capability. Moreover, the effects of key control parameters on multiple performance indices are analyzed to provide references for further research on combined energy and attitude control systems.