I would like to offer a gedanken approach to that question.
Imagine you are at a stationary system (the lab frame) $S$ in which time is discrete, with regard to some smallest increment of time ($Δt=κ$, a "chronon" if you will).
Within that frame, the maximal acceleration possible, $α$, is given by the maximal change in velocity (from going at the speed of light $c$ in the positive direction to going at $c$ in the negative direction- $Δv=2c$) over the minimal duration in which it can happen (a single chronon- $Δt=κ$). We get:
(In a similiar manner maximal jerk, pop etc. can be derived)
I am not quite sure how I should regard $α$. A maximal acceleration with respect to a frame sounds a bit off to me. If it is identical in all frames we recieve something like one of the the postulates of special relativity- which might sprout higher orders of special relativity.
If it is not identical for all frames, then I am not even sure how it can be treated. Note that this is not a maximal acceleration in the object's own rest frame (like the Schwinger limit), but a frame-dependant limit.
Another approach I can think of is that in a discrete-time world, time translation symmetry becomes discrete time translation symmetry, which means conservation of energy is replaced by some discrete time parallel, which I am not familiar with. Maybe nonconsevations of energy can be used to suggest for or against the idea of discrete time.
If anyone could help me continue one of the two lines of thought introduced here, I think discrete time can lead to a contradiction, but I can't quite get there myself.