Efficient SARS-CoV-2 (COVID-19) detection is critical, yet RT-PCR testing remains time-consuming and costly. Chest X-ray imaging provides a faster, more accessible alternative. This study introduces DLR-CovNet, a dual-scale lightweight deep learning model with 2.22M parameters, designed for accurate CXR-based diagnosis in resource-constrained settings. The study employs the COVIDx CXR-4 dataset, which suffers from significant class imbalance, addressed through the Inverse Class Frequency Re-weighting method. DLR-CovNet incorporates a recursively applied Residual Dual-Conv Pooling (RDCP) block that enables dual-scale feature learning through varying kernel sizes. The RDCP block integrates residual connections, batch normalization, and dual sequential convolution layers, enhancing feature representation and classification accuracy. Outperforming well-known deep learning architectures, the model achieves 93% accuracy, 92% precision, 93% recall, 93% F1-score, and an MCC (Matthews Correlation Coefficient) score of 85%. Additionally, Grad-CAM is employed to analyze model interpretability by visualizing regions of interest in CXR images for SARS-CoV-2 detection.