Here's a way to avoid it: Change the process. Do counting, re-counting, appeals, etc. first, until a final decision as to the correct count and valid votes is made (all appeals exhausted or no longer possible due to time elapsed). We have final decisions on all these matters in current systems. This way, you will "know" that the election was fair, since this depends on the ballots, not the candidate.

Then do the random step. The drawback is that there may be a month or two between casting votes and having the results, but I don't see why this is a big deal.