I think one can justify using continuous methods without depending on whether the universe itself is made of continuous structures. If everything is imagined to be discrete, but you don’t know the scale of the discreteness other than it being much smaller than anything we can observe, then smooth manifolds are the way to formalise that. Even where we can observe the discreteness, such as fluids being made of atoms, the continuous model still works for most practical purposes, especially when augmented with perturbation methods when the mean free path is of a small but macroscopic size. No ontological commitment to continuity is needed. Continuity is instead the way to think about indefinitely fine division.
I think one can justify using continuous methods without depending on whether the universe itself is made of continuous structures. If everything is imagined to be discrete, but you don’t know the scale of the discreteness other than it being much smaller than anything we can observe, then smooth manifolds are the way to formalise that. Even where we can observe the discreteness, such as fluids being made of atoms, the continuous model still works for most practical purposes, especially when augmented with perturbation methods when the mean free path is of a small but macroscopic size. No ontological commitment to continuity is needed. Continuity is instead the way to think about indefinitely fine division.