In converter-dominated direct-current microgrids, severe load transients can cause large voltage deviations on the common direct-current bus. To mitigate this, an energy storage system is typically employed, and an isolated bidirectional dual active bridge converter is commonly used as the power interface. Therefore, the controller must ensure robust transient performance under step-load conditions. This paper proposes an active disturbance rejection control framework that adaptively adjusts the bandwidth of an extended state observer using fuzzy logic. The proposed observer increases its bandwidth during transients—based on the estimation error—to accelerate disturbance compensation, while decreasing the bandwidth near steady state to suppress noise amplification. This adaptive tuning alleviates the fixed-bandwidth trade-off between transient speed and noise sensitivity in ESO-based regulation. Hardware experiments under load-step conditions validate the method: for a load increase, the peak voltage undershoot and settling time are reduced by 22% and 48.9% relative to a proportional–integral controller, and by 20% and 36.1% relative to a fixed-bandwidth observer. For a load decrease, the peak overshoot and settling time are reduced by 27.9% and 49.5% compared with the proportional–integral controller, and by 20.5% and 25% compared with the fixed-bandwidth observer.