If we could take a snapshot of both tennis balls, would there be any evidence that could suggest that one is moving and the other one is still?
We can't. Problem solved.
Well, almost problem solved. So in reality, we can take shorter and shorter exposures. I can take a 1 second exposure of the scene, where the moving tennis ball will be heavily blurred while the stationary one will be crisp. I can capture the same scene at 1/100th of a second, and moving ball will look more crisp like the stationary one. I can capture the same scene at 1/1000th of a second, and it will be very difficult for the human eye to discern which one is in motion. I can make these snapshots shorter and shorter. Indeed, we have looked at imaging scenes at such exacting shutter speeds that we can watch light propagate through the scene. But we never quite hit a perfect standstill. We never hit an infinitely fast shutter speed.
Now forgive me if I handwave a bit, but there is an unimaginably large body of evidence that motion exists. In particular, you'll fail to predict very much if you assume no motion occurs. So from that empirical point of view, we should find that motion exists. From a philosophical point of view, there's some interesting questions to be had regarding endurable versus perdurable views of the universe, but from a scientific perspective, we generally agree that motion exists.
So how do we resolve the conundrum you are considering? The answer is calculus. Roughly 400 years ago, Isaac Newton and Gottfried Leibniz independently developed a consistent way of dealing with infinitesimally small values. We generally accept this as the "correct" way of handling them. It does not permit us to consider a shutter speed which is truly infinite, letting us isolate a moment perfectly, to see if there is motion or not, but it does let us answer the question "what happens if we crank the shutter speed up? What if we go 1/100th of a second, 1/1000th, 1/100000th, 1/0000000000th of a second and just keep going?" What happens if we have an infinitesimally small exposure period in our camera?
Using that rigor, what we find is that modeling the world around us really requires two things. The first is the values you are familiar with, such as position. And the second is the derivatives of those familiar things, such as velocity. These are the results of applying the calculus to the former group.
We find that models such as Lagrangian and Hamiltonian models of systems work remarkably well for predicting virtually all systems. These systems explicitly include this concept of a derivative in them, this idea of an "instantaneous rate of change." So we say there is motion, because it seems unimaginably difficult to believe that these patterns work so well if there was not motion!
As a side note, you set up your experiment in space, so there's nothing much to interact with. However, had you set the experiment up in the water, you would find the chaotic flow behind the moving ball very interesting. It would be ripe with fascinating and beautiful twirls that are very hard to explain unless associated with some forward motion.