Are there any quantum theories where particles have discrete probability distributions?

I have done a bit of searching for an answer to this question, but I am an amateur and suspect I lack the proper "language" to describe what I’m actually looking for, so a pointer in the right direction would be much appreciated!

Let’s presume that spacetime is in fact quantized (how or why we don’t care), and we have a field overlayed on top of it. We emit a particle and take a measurement, and determine its precise location. Due to the quantized nature of spacetime by our assumption, we would expect that the measured position would occur only at the lattice points of spacetime itself. This seems to imply (in my mind) that a theory of quantum gravity would necessitate that the probably density of the wavefunction itself would have to be discrete as there would be forbidden values for location.

So my questions are, first, does that even make sense/matter? Is there some underlying property of a field that would allow it to take on a continuous probability distribution even if spacetime is quantized? And second, if one would expect that the probability density function would be discrete, has anyone attempted to start out with that presumption and work backwards toward a theory of quantum gravity?