Real-world driving involves intricate interactions among vehicles navigating through dense traffic scenarios. Recent research focuses on enhancing the interaction awareness of autonomous vehicles to leverage these interactions in decision-making. These interaction-aware planners rely on neural-network-based prediction models to capture inter-vehicle interactions, aiming to integrate these predictions with traditional control techniques such as Model Predictive Control. However, this integration of deep learning-based models with traditional control paradigms often results in computationally demanding optimization problems, relying on heuristic methods. This study introduces a principled and efficient method for combining deep learning with constrained optimization, employing knowledge distillation to train smaller and more efficient networks, thereby mitigating complexity. We demonstrate that these refined networks maintain the problem-solving efficacy of larger models while significantly accelerating optimization. Specifically, in the domain of interaction-aware trajectory planning for autonomous vehicles, we illustrate that training a smaller prediction network using knowledge distillation speeds up optimization without sacrificing accuracy.