I just thought of a way to possibly make it more intuitive. Use ranked voting, with the addition of a “nobody below this line” item that you can add to your ranking anywhere you like. Then at the runoff stage, you use approval voting, with “nobody below here” determining which candidates you approve.
That gives you a nice easy-to-understand runoff, and is maybe even more understandable than normal ranked voting because you get to explicitly separate out the candidates you really can’t stomach. I suspect some people don’t like ranked voting because you’re expected to stack-rank lots of candidates you dislike.
But we're talking a lot of complexity to have both ranks and scores as separate marks. Super long ballot, will turn away most voters and seem really hard to process.
It sounds like you might appreciate 3-2-1 voting. https://wiki.electorama.com/wiki/3-2-1_voting
(The point of the above was to respect the Condorcet Criterion, and 3-2-1 voting does not meet that criterion.)
So one option is to hold a runoff between the candidates in the Smith Set.
Another is crazier, but I haven't found a flaw yet. Basically introduce range/score voting on the ballot, but only use the scores if there is a multi-member Smith Set. That way you guarantee Condorcet compliance. And since the voters don't know if the ranges/scores will be used (and they probably won't), they don't have incentive to strategically score a less-desired candidate ahead of a more-desired candidate.