Chest X-ray (CXR) interpretation is a fundamental yet complex clinical task that increasingly relies on artificial intelligence for automation. However, traditional monolithic models often lack the nuanced reasoning required for trustworthy diagnosis, frequently leading to logical inconsistencies and diagnostic hallucinations. While multi-agent systems offer a potential solution by simulating collaborative consultations, existing frameworks remain susceptible to consensus-based errors when instantiated by a single underlying model. This paper introduces XrayClaw, a novel framework that operationalizes multi-agent alignment through a sophisticated cooperative-competitive architecture. XrayClaw integrates four specialized cooperative agents to simulate a systematic clinical workflow, alongside a competitive agent that serves as an independent auditor. To reconcile these distinct diagnostic pathways, we propose Competitive Preference Optimization, a learning objective that penalizes illogical reasoning by enforcing mutual verification between analytical and holistic interpretations. Extensive empirical evaluations on the MS-CXR-T, MIMIC-CXR, and CheXbench benchmarks demonstrate that XrayClaw achieves state-of-the-art performance in diagnostic accuracy, clinical reasoning fidelity, and zero-shot domain generalization. Our results indicate that XrayClaw effectively mitigates cumulative hallucinations and enhances the overall reliability of automated CXR diagnosis, establishing a new paradigm for trustworthy medical imaging analysis.