This paper presents a hardware-efficient deep neural network (DNN), optimized through hardware-aware neural architecture search (HW-NAS); the DNN supports the classification of session-level encrypted traffic on resource-constrained Internet of Things (IoT) and edge devices. Thanks to HW-NAS, a 1D convolutional neural network (CNN) is tailored on the ISCX VPN-nonVPN dataset to meet strict memory and computational limits while achieving robust performance. The optimized model attains an accuracy of 96.59% with just 88.26K parameters, 10.08M FLOPs, and a maximum tensor size of 20.12K. Compared to state-of-the-art models, it achieves reductions of up to 444-fold, 312-fold, and 15.6-fold in these metrics, respectively, significantly minimizing memory footprint and runtime requirements. The model also demonstrates versatility in classification tasks, achieving accuracies of up to 99.64% in VPN differentiation, VPN-type classification, broader traffic categories, and application identification. In addition, an in-depth approach to header-level preprocessing strategies confirms that the optimized model can provide notable performances across a wide range of configurations, even in scenarios with stricter privacy considerations. Likewise, a reduction in the length of sessions of up to 75% yields significant improvements in efficiency, while maintaining high accuracy with only a negligible drop of 1-2%. However, the importance of careful preprocessing and session length selection in the classification of raw traffic data is still present, as improper settings or aggressive reductions can bring about a 7% reduction in overall accuracy. Those results highlight the method's effectiveness in enforcing cybersecurity for IoT networks, by providing scalable, efficient solutions for the real-time analysis of encrypted traffic within strict hardware limitations.