Fixed-point actions are classically perfect lattice actions, i.e., they are free from classical lattice artifacts. Furthermore, they exhibit suppressed quantum artifacts. Monte Carlo simulations employing such actions may provide a way to efficiently approach the continuum limit on coarse lattices, thereby avoiding critical slowing down and topological freezing. Extending our previous work, we use lattice gauge equivariant convolutional neural networks (L-CNNs) to approximate a fixed-point action for SU(3) gauge theory in four dimensions to previously unseen accuracy. Using this new parametrization, we perform HMC simulations and classically perfect gradient flow. Our self-consistent approach aims to extract gradient flow observables on much coarser lattices compared to simulations using the Wilson action.