With its dual-functional advantages, integrated sensing and communications (ISAC) technologies can be further extended to satellite communications, enhancing global coverage services. However, achieving vast coverage would result in significant delays and considerable path losses. Motivated by this, in this paper, we focus on satellite-based ISAC (S-ISAC) systems and propose a general transceiver design framework incorporating both transmit waveform and receive filter. Unlike existing approaches, our approach uses a predictive joint transmit waveform and receive filter design that eliminates the need of channel estimation, thereby reducing time overhead. Additionally, a versatile weighting mechanism is designed to allow flexible prioritization between communications and sensing. To tackle the intractability of the ISAC transceiver design problem, we adopt a data-driven deep learning-based approach, where the model learns to design the transmit waveform and receive filter from historical channel data. Specifically, we propose a predictive optimization network (PONet), leveraging convolutional layers and a Transformer encoder to capture long-term spatial-temporal features and facilitate the learning capability. Numerical results demonstrate the effectiveness of the proposed PONet in terms of communications and sensing rates in S-ISAC networks in various system settings.