How do you prove an EV battery capacity is below a certain percentage?

You could certainly have the vehicle’s battery packs tested for efficiency by an independent test centre. But why not keep that option up your sleeve and let the dealership test it initially? I would imagine the battery’s performance and efficiency test would involve a written report which is much more difficult to fudge than a verbal report. If the dealership can’t provide this written proof on the battery’s condition and you still believe you have a problem, that’s when you play the independent-test card.