The sympathetic nervous system (SNS) plays a central role in regulating the body's responses to stress and maintaining physiological stability. Its dysregulation is associated with a wide range of conditions, from cardiovascular disease to anxiety disorders. Skin nerve activity (SKNA) extracted from high-frequency electrocardiogram (ECG) recordings provides a noninvasive window into SNS dynamics, but its measurement is highly susceptible to electromyographic (EMG) contamination. Traditional preprocessing based on bandpass filtering within a fixed range (e.g., 500--1000 Hz) is susceptible to overlapping EMG and SKNA spectral components, especially during sustained muscle activity. We present a denoising approach using a lightweight one-dimensional convolutional autoencoder with a long short-term memory (LSTM) bottleneck to reconstruct clean SKNA from EMG-contaminated recordings. Using clean ECG-derived SKNA data from cognitive stress experiments and EMG noise from chaotic muscle stimulation recordings, we simulated contamination at realistic noise levels (--4 dB, --8 dB signal-to-noise ratio) and trained the model in the leave-one-subject-out cross-validation framework. The method improved signal-to-noise ratio by up to 9.65 dB, increased cross correlation with clean SKNA from 0.40 to 0.72, and restored burst-based SKNA features to near-clean discriminability (AUROC $\geq$ 0.96). Classification of baseline versus sympathetic stimulation (cognitive stress) conditions reached accuracies of 91--98\% across severe noise levels, comparable to clean data. These results demonstrate that deep learning--based reconstruction can preserve physiologically relevant sympathetic bursts during substantial EMG interference, enabling more robust SKNA monitoring in naturalistic, movement-rich environments.