Deep-learning models can achieve strong chest X-ray (CXR) classification performance without establishing whether their predictions predominantly rely on pulmonary image content. This study evaluates pulmonary attribution containment as an anatomy-related reliability property distinct from diagnostic performance. We propose DBCA-SegNet-MGAP, a multi-task anatomy-guided CNN-Transformer framework that combines complementary feature representations through bidirectional cross-backbone attention, predicts a soft lung mask, and incorporates this anatomical prior directly into classification through Mask-Guided Adaptive Global Average Pooling (MGAP). Pulmonary attribution containment is quantified using the Anatomical Local Energy Ratio (ALR) and high-intensity cumulative ALR ([email protected]). Experiments were repeated across three training seeds using the COVID-19 Radiography Database for four-class internal testing and a locked Shenzhen-to-Montgomery protocol for zero-shot external tuberculosis testing. On COVID-19, the proposed model achieved a weighted F1 of $0.9615 \pm 0.0015$ and macro ROC-AUC of $0.9906 \pm 0.0007$. In an architecture-matched dual-bridge comparison, replacing conventional GAP with MGAP increased ALR from $0.3878 \pm 0.0098$ to $0.7086 \pm 0.0104$ and [email protected] from $0.5265 \pm 0.0101$ to $0.9905 \pm 0.0018$, while weighted F1 remained essentially unchanged ($0.9618 \pm 0.0015$ vs. $0.9615 \pm 0.0015$). Under locked external transfer to Montgomery, ROC-AUC remained $0.9080 \pm 0.0043$ and pulmonary ALR remained $0.6466 \pm 0.0081$, whereas weighted F1 decreased to $0.7528 \pm 0.0080$ and ECE increased to $0.1683 \pm 0.0055$. These findings show that diagnostic discrimination, calibration, and pulmonary attribution containment are distinct model properties and support their joint evaluation under internal testing and external domain shift.

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