If you want a falsifiable theory you must make a prediction about specific mutually exclusive ways the universe could be versus not be and then argue that one the groups must happen or cannot happen. Then when you investigate and find which one you get, you have falsified or not falsified the theory.
So ask yourself what you predictions are. Your predictions are a bit vague, but they sound like the predictions of a continuous theory. Why do I say that. It's because of the "at some level" part. It sounds like it means that at scales much larger than that hypothetical level every prediction agrees with the continuous theory.
So you could make the same predictions as a continuous theory and then whenever data is collected, no matter what data we see, the data was collected at some scale and you could just say that if only the scale was smaller things would have turned out differently.
But however small the scale is, you can pull the same trick. You never even need to bother making discrete predictions because whatever data fits the continuous theory also allows the discrete one to exist at the much smaller resolution without being exposed.
This freedom to wiggle out of any data is the hallmark of an unfalsifiable theory.
But if you claimed there was a discrete theory at a fixed level where experiments with a particular nonzero scale were small enough to require different predictions for the discrete theory. Now you have made a prediction that can be tested and thus your theory is falsifiable.
So you could have a whole family of theories, each predicting deviations at a different scale. And each one would be falsifiable. But the meta claim that at least one of them is correct, that meta claim is not falsifiable.