Federated Learning (FL) is a privacy-preserving distributed learning paradigm designed to build a highly accurate global model. In Mobile Edge IoT (MEIoT), the training and communication processes can significantly deplete the limited battery resources of devices. Existing research primarily focuses on reducing overall energy consumption, but this may inadvertently create energy consumption imbalances, leading to the premature dropout of energy-sensitive devices.To address these challenges, we propose BEFL, a joint optimization framework aimed at balancing three objectives: enhancing global model accuracy, minimizing total energy consumption, and reducing energy usage disparities among devices. First, taking into account the communication constraints of MEIoT and the heterogeneity of devices, we employed the Sequential Least Squares Programming (SLSQP) algorithm for the rational allocation of communication resources. Based on this, we introduce a heuristic client selection algorithm that combines cluster partitioning with utility-driven approaches to alleviate both the total energy consumption of all devices and the discrepancies in energy usage.Furthermore, we utilize the proposed heuristic client selection algorithm as a template for offline imitation learning during pre-training, while adopting a ranking-based reinforcement learning approach online to further boost training efficiency. Our experiments reveal that BEFL improves global model accuracy by 1.6\%, reduces energy consumption variance by 72.7\%, and lowers total energy consumption by 28.2\% compared to existing methods. The relevant code can be found at \href{URL}{https://github.com/juzehao/BEFL}.