Satellite SAR (synthetic aperture radar) imagery offers global coverage and all-weather recording capabilities, making it valuable for applications like remote sensing and maritime surveillance. However, its use in machine learning-based automatic target classification faces challenges, including the limited availability of SAR target training samples and the inherent constraints of SAR images, which provide less detailed features compared to natural images. These issues hinder the effective training of convolutional neural networks (CNNs) and complicate the transfer learning process due to the distinct imaging mechanisms of SAR and natural images. To address these challenges, we propose a shallow CNN architecture specifically designed to optimize performance on SAR datasets. Evaluations were performed on three datasets: FUSAR-Ship, OpenSARShip, and MSTAR. While the FUSAR-Ship and OpenSARShip datasets present difficulties due to their limited and imbalanced class distributions, MSTAR serves as a benchmark with balanced classes. To compare and optimize the proposed shallow architecture, we examine various properties of CNN components, such as the filter numbers and sizes in the convolution layers, to reduce redundancy, improve discrimination capability, and decrease network size and learning time. In the second phase of this paper, we combine the CNN with Long short-term memory (LSTM) networks to enhance SAR image classification. Comparative experiments with six state-of-the-art CNN architectures (VGG16, ResNet50, Xception, DenseNet121, EfficientNetB0, and MobileNetV2) demonstrate the superiority of the proposed approach, achieving competitive accuracy while significantly reducing training times and network complexity. This study underscores the potential of customized architectures to address SAR-specific challenges and enhance the efficiency of target classification.