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I claim that the results of the calorimetry may be due to the energy of
the positive atoms, not the free electrons. So if you combine the
results of the "current experiment" that you described, which only gives
the energy of the free electrons, with the calorimetry experiment, you
get 1/2 CV^2 + 1/2 CV^2 = CV^2 for the total energy stored in the
capacitor.
I'm not claiming that either one of the experiments, alone, will give
CV^2, but why couldn't you combine both experiments together in the same
experiment? That is, do the calorimetry experiment and the current
experiment at the same time, using the same circuit, and combine the
results of the two measurements.