Let's say you had an EM wave from a distant source hitting a parallel plate capacitor as in the above setup. Would the battery "hold" the voltage between the plates constant, thus holding the EM field constant between the plates (effectively blocking the wave between the plates), or would the wave just add to the existing field between the plates (superposition principle)?
Let's symplify your situation. Let's assume that we have charged capasitor in equillibrium with some certain charges on it and certain field (black arrows) in it
Now imagine that we add resting charge in some distance and it violates the equillibrium by his own field (red arrows)
Now the field in the capasitor is a bit bigger that makes the potential difference a bit bigger, that violates the equillibrium between "circuit" and source. This small difference in voltage will just manifest itself as moving force for charges in the "circuit" that will change the amount of charge in capacitor, the current will appear for short time before the system will come into new equillibrium with smaller charge. The difference in voltages will lay on the wires between capacitor and source.
As a result we can make a conclusion that system will always change itself to compensate voltage conditions or strong current will appear and compensate it anyway. Your case with the waves can be described just like oscillating distant charge, that will cause oscillating current in the capasitor. If oscillating is slow enough the current will be on time to compensate the difference between source and field, otherwise potential difference would also oscillate.