Immersive extended reality (XR) applications introduce latency-critical workloads that must satisfy stringent real-time responsiveness while operating on energy- and battery-constrained devices, making execution placement between end devices and nearby edge servers a fundamental systems challenge. Existing approaches to adaptive execution and computation offloading typically optimize average performance metrics and do not fully capture the sustained interaction between real-time latency requirements and device battery lifetime in closed-loop XR workloads. In this paper, we present a battery-aware execution management framework for edge-assisted XR systems that jointly considers execution placement, workload quality, latency requirements, and battery dynamics. We design an online decision mechanism based on a lightweight deep reinforcement learning policy that continuously adapts execution decisions under dynamic network conditions while maintaining high motion-to-photon latency compliance. Experimental results show that the proposed approach extends the projected device battery lifetime by up to 163% compared to latency-optimal local execution while maintaining over 90% motion-to-photon latency compliance under stable network conditions. Such compliance does not fall below 80% even under significantly limited network bandwidth availability, thereby demonstrating the effectiveness of explicitly managing latency-energy trade-offs in immersive XR systems.