Reducing preventable maternal mortality remains a global health priority. Under Sustainable Development Goal (SDG) target 3.1, the WHO emphasizes timely and equitable allocation of limited maternal health resources. Motivated by Department of Obstetrics and Gynecology at several important hospitals in Uganda and Ghana, we study the problem of sequential allocation of wearable vital sign monitoring devices among maternal mothers. While deep reinforcement learning (RL) has shown promise for sequential resource allocation, its limited interpretability hinders adoption in such high-stakes settings. In contrast, large language model (LLM) agents provide human-readable reasoning but often struggle with effective long-term decision making. To bridge this gap, we introduce Rule-Bottleneck RL (RBRL), the first LLM agent framework for resource allocation problems that jointly optimizes language-based decision policy and explainability. At each step within RBRL, an LLM first generates candidate rules---language statements capturing decision priorities tailored to the current state. RL then optimizes rule selection to maximize environmental rewards and explainability, with the LLM acting as a judge. Finally, an LLM chooses the action (optimal allocation) based on the rule. We provide conditions for RBRL performance guarantees as well as the finite-horizon evaluation gap of the learned RBRL policy. Experiments in maternal health show that RBRL outperforms baseline LLM agents and approaches the performance of deep RL, while producing clearer, policy-relevant explanations. Human evaluations further confirm improved trust and usability, demonstrating RBRL as a practical AI approach aligned with SDG target 3.1.