Frustratingly, attempts to discretize space invariably run into problems with relativity, since they effectively impose a preferred frame of reference. I.e. you can impose a minimum distance, but relativistic length contraction means that observers measure different minima and in different directions.

Apparently, under some of these models, this implies that the speed of light ends up depending on wavelength, lending them to empirical tests. My understanding is that these discrete space models have failed to line up with experiment, at least within the limits of measurement.

That's currently unknown. For all current practical purposes, kind of. The plank length sets a limit on spatial resolution of any information, so a finite region with (universally) bounded entropy per conceivable bucket on that scale still has finite entropic capacity.

The current theories use continuous space and time. However, we can't encode information into space itself. We would have to use some configuration of matter, and then there are limits to how well-defined a particle's position can be coming from the uncertainty principle.

On the other hand, general relativity implies that if you put enough matter in a small enough space it becomes a black hole, and then we can't access the information in it.

IANA physicist but I think this line of thought is probably speculative at the moment because it involves both general relativity and quantum mechanics and it isn't known how they should work together.