Tuberculosis (TB) remains a major global health threat, particularly in low-resource settings where timely diagnosis is critical yet often limited by the lack of radiological expertise. Chest X-rays (CXRs) are widely used for TB screening, but manual interpretation is prone to errors and variability. While deep learning has shown promise in automating CXR analysis, most existing models are computationally intensive and lack interpretability, limiting their deployment in real-world clinical environments. To address this gap, we evaluated three lightweight and explainable CNN architectures, ShuffleNetV2, SqueezeNet 1.1, and MobileNetV3, for binary TB classification using a locally sourced dataset of 3,008 CXR images. Using transfer learning and Grad-CAM for visual explanation, we show that MobileNetV3 and ShuffleNetV2 achieved perfect test performance with 100% accuracy, sensitivity, specificity, precision, and F1-score, along with AUC scores of 1.00 and inference times of 94.66 and 103.63 seconds, respectively. SqueezeNet performed moderately, with a lower F1-score of 82.98% and several misclassifications. These results demonstrate that lightweight CNNs can deliver high diagnostic accuracy and transparency, supporting their use in scalable, AI-assisted TB screening systems for underserved healthcare settings.