Pilot readback of Air Traffic Control (ATC) voice instructions is a primary safeguard against miscommunication in air transportation. However, readback anomalies remain implicated in approximately 80% of aviation incidents. This vulnerability is further exacerbated by rising traffic volume and elevated cognitive workload, thereby motivating automated readback monitoring by machine. Traditional rule-based and machine learning approaches struggle to generalize across the highly variable and evolving phraseology of air traffic controller-pilot communications. While Large Language Models (LLMs) have opened a new avenue through their strong reasoning and generalization capabilities, existing approaches still face deployment and computational barriers in practice. In this work, we propose Semantic reasoning for Communication via Open-set Plug-in with Examples (SCOPE), a novel lightweight-training LLM framework that advances both the efficiency and accuracy of machine-based ATC readback monitoring. The core idea is to couple a plug-in open-set classifier with a carefully designed in-context learning mechanism on top of a frozen LLM. Extensive experiments on the semi-synthetic communication dataset show that SCOPE attains superior accuracy while delivering the low-latency response required for operational environments. Under a few-shot setting, SCOPE achieves 91.05% accuracy in open-set detection and corrects 96.63% of anomalous readbacks, thereby outperforming the strongest available baselines while providing explanations for its decisions. These findings demonstrate the potential of our framework as a practical pathway toward interpretable and controllable ATC readback monitoring.