Pneumonia remains a leading cause of childhood mortality worldwide, with a heavy burden in low-resource settings such as Bangladesh where radiologist availability is limited. Most existing deep learning approaches treat pneumonia detection as a binary problem, overlooking the clinically critical distinction between bacterial and viral aetiology. This paper proposes CBAM-DenseNet121, a transfer-learning framework that integrates the Convolutional Block Attention Module (CBAM) into DenseNet121 for three-class chest X-ray classification: Normal, Bacterial Pneumonia, and Viral Pneumonia. We also conduct a systematic binary-task baseline study revealing that EfficientNetB3 (73.88%) underperforms even the custom CNN baseline (78.53%) -- a practically important negative finding for medical imaging model selection. To ensure statistical reliability, all experiments were repeated three times with independent random seeds (42, 7, 123), and results are reported as mean +/- standard deviation. CBAM-DenseNet121 achieves 84.29% +/- 1.14% test accuracy with per-class AUC scores of 0.9565 +/- 0.0010, 0.9610 +/- 0.0014, and 0.9187 +/- 0.0037 for bacterial pneumonia, normal, and viral pneumonia respectively. Grad-CAM visualizations confirm that the model attends to anatomically plausible pulmonary regions for each class, supporting interpretable deployment in resource-constrained clinical environments.