Model Predictive Control (MPC) is effective at generating safe control strategies in constrained scenarios, at the cost of computational complexity. This is especially the case in robots that require high sampling rates and have limited computing resources. Differentiable Predictive Control (DPC) trains offline a neural network approximation of the parametric MPC problem leading to computationally efficient online control laws at the cost of losing safety guarantees. DPC requires a differentiable model, and performs poorly when poorly conditioned. In this paper we propose a system decomposition technique based on relative degree to overcome this. We also develop a novel safe set generation technique based on the DPC training dataset and a novel event-triggered predictive safety filter which promotes convergence towards the safe set. Our empirical results on a quadcopter demonstrate that the DPC control laws have comparable performance to the state-of-the-art MPC whilst having up to three orders of magnitude reduction in computation time and satisfy safety requirements in a scenario that DPC was not trained on.