Deep learning (DL)-based methods in pre-fault dynamic security assessment (DSA) have provided significant results, contributing to the safe operation of power systems. However, power systems often suffer from insufficient, small, and imbalanced datasets, which significantly impact the performance of DL-based DSA models. Existing DSA frameworks typically operate as two-class black-box models, assessing only overall system security without providing insights into the causes of insecurity or identifying critical generators (CGs), and they fail to quantify prediction uncertainty. These challenges hinder the implementation of current methods in real-world power systems and reduce operators' confidence in them. To address these issues, this paper proposes an uncertainty-aware bi-level multitask learning framework based on transfer learning and SqueezeNet architecture. The framework assesses system security, identifies CGs during instability, and leverages fine-tuning of a pre-trained SqueezeNet model to facilitate training with limited data. Additionally, evidential deep learning is incorporated to quantify classification uncertainty. Without relying on the complex and challenging data augmentation method, this framework uses a simple technique called optimal classification threshold determination to mitigate the negative impact of imbalanced data on model performance. The optimal threshold is determined by maximizing the area under the receiver operating characteristic (ROC) curve. The application of the proposed method to the IEEE 118-bus system shows its strong performance. These results offer crucial technical insights for the implementation of DL-based DSA in real-world power systems.