Discretization is not really the problem: you instead have to fix the definite position and momentum of the microscopic components of the square such that all the parts of it stay in place relative to one another and to the total arrangement of the shape they collectively form.

Switching to a taxicab geometry doesn't change the tendency of matter to move around; it only defines the points in which matter might be found in principle, and where it can never be found, ever.

In effect, the dynamism of spacetime is a red herring (we have difficulties building ideal spheres in flat space, theoretically speaking), and spacetime geometry (in the sense of minimal lengths vs infinite differentiability) is at best a marginal contribution to the problem.

While Lorentz invariance is essentially what blocks the quantization of energy, you can still have a discretized spacetime that preserves Lorentz invariance. One broad approach to that involves a Lorentz contraction on the spacings between the allowed points, so that observers may disagree on the lengths of objects marked off in Cartesian coordinates on the fundamental lattice spacings. In particular, even in the absence of gravitation, one observer will conclude a small object has a definite lattice length while a generic accelerated observer (and at least some boosted observers) will conclude that the same small object's length is in superposition.

Additionally, one may quantize lengths and times as in a Snyder spacetime, and still recover Lorentz invariance (or at the very least do away with preferred frames; things are tricky where boosts are high and ultimately it is likely impossible to recover the Lorentz subgroup in its entirety) in a quite natural fit with a deformation of Special Relativity.

Unfortunately we still have GR with its strong successes and several viable models which have minimum length scales. So at least at the energy limits we have access to now, we can't really make a concrete choice about which better describes our universe. In part that's because almost all these minimal-length theories are carefully designed to reproduce General Relativity in some limit because of its strong successes.