Long-duration power outages caused by natural disasters can severely disrupt society, leading to widespread social and economic losses. Although such outages can be considered a type of disaster, the response to them is not typically coordinated with other disaster response activities. Previous work has overlooked the opportunities that arise from enhanced coordination between evacuation logistics and backup power support. To fill this gap, this paper develops an integrated framework of power outage disaster response that coordinates localized battery-based power restoration with emergency evacuation and prioritizes support to socially vulnerable populations who are most at risk during an outage. The proposed sequential optimization models generate an evacuation collection schedule that informs temporary power grid restoration with mobile energy storage in an isolated distribution system. The model is demonstrated through a case study of the city of Greeley, Colorado during a flooding event. Sensitivity analyses examine the impact of evacuation compliance uncertainty and road network congestion on the modeling results, demonstrating the robustness of the framework to uncertainty and the benefits of mobile resources over stationary counterparts. Results indicate that evacuation-informed localized power restoration can provide additional support for vulnerable residents who need it the most without compromising the overall power restoration performance.