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Dmytry comments on Can You Prove Two Particles Are Identical? - Less Wrong
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How exactly does something not distinguish E? In your charge example, wouldn't they interact differently with the electric field that would always be present?
For some things it might make no difference in the limit. That is, as the electric field decreases the results of the experiment approach protons and neutrons seeming to be identical, but that isn't true about this experiment. You're twice as likely to observe the same electron twice than two with masses that differ in their first decimal point. You're twice as likely to observe the same electron twice than two with masses that differ in their second decimal point. In general, you're twice as likely to observe the same electron twice than two with masses that differ in their nth decimal point. The limit as the electrons approach the same mass is that the experiment still distinguishes them.
They would interact differently with electric field, but it takes time until there is interaction.
With the electron masses - not sure how you'd go about masses but if you had some label on the electron, which does not interact with anything, and assuming MWI, you would get the statistics as if they are identical. If the label interacts with something, one has to make it interact to make it as if they are distinguishable.