> If spacetime had a discrete character at scales like the inverse of the universe scale we would see dispersion of light as it traveled cosmological distances and we do not observe this. It is technically possible that the discreteness scale is much, much smaller than the inverse universe scale, of course, but at this point it seems pointless to me to entertain discrete models

A computational universe does not strictly imply discrete spacetime. You can most certainly still have a continuous universe—at least from the perspective of the beings that inhabit it. By way of analogy, consider the fact that ZFC proves the existence of uncomputable real numbers yet itself has a countable model (presuming it is consistent).