The part of the impossibility proof that deals with dictators is subtle and neat. At first, I was convinced that the dictator was actually a "swing voter" whose power was entirely conditioned on the other voter's preferences, but that's not actually the case.

Still, my guess is that these types of edge cases could be handled by using some randomness and low probability arguments. For example, claiming that such dictators exist with negligible probability given a small-ish fraction of voters that are considered "random."

While you would never want to inject random votes into an election, it's probably reasonable to assume that some fraction of voters behave naturally random (e.g., people get sick and don't vote, or are just politically fickle). This probably conflicts with the property of "unrestricted domain" but it seems like a reasonable concession in practice.