In recent years, deep learning has been widely used to solve the bottleneck problem of synthetic aperture radar (SAR) automatic target recognition (ATR). However, most current methods rely heavily on a large number of training samples and have many parameters which lead to failure under limited training samples. In practical applications, the SAR ATR method needs not only superior performance under limited training data but also real-time performance. Therefore, we try to use a lightweight network for SAR ATR under limited training samples, which has fewer parameters, less computational effort, and shorter inference time than normal networks. At the same time, the lightweight network combines the advantages of existing lightweight networks and uses a combination of MnasNet and NetAdapt algorithms to find the optimal neural network architecture for a given problem. Through experiments and comparisons under the moving and stationary target acquisition and recognition (MSTAR) dataset, the lightweight network is validated to have excellent recognition performance for SAR ATR on limited training samples and be very computationally small, reflecting the great potential of this network structure for practical applications.