Consider the following method, which I’m calling Approval vs. B2R, similar to Approval-seeded Llull as per @Jack-Waugh:
(1) Voters submit ranked ballots with an approval cutoff.
(2) Candidates are sorted by approval rating, with ties solved by head-to-head when possible.
(3) The Bottom Two Runoff (B2R) winner is computed with scores set as approval rates, with head-to-head ties solved by approval rates of possible.
(4) Of the non-B2R candidates with the highest approval rating, a contender is chosen (by head-to-head selection if possible, else by tie break).
(5) The B2R winner and the Approval contender go against each other in a final, independent head-to-head, with important but possibly strange-seeming caveats: the Approval winner’s advantage is increased by +1, and ties go to the Approval contender.
I claim that these caveats about the runoff—including setting aside the B2R winner, increasing the Approval contender’s advantage by +1, and giving them ties—together restore the participation criterion under fully sincere ballots, at the expense of the full Condorcet criterion. However, when the margins are not close, the Condorcet criterion is still satisfied if ballots are sincere.
This may cause strange effects at the edge cases, such as when the approval and Condorcet winner coincide, but the Condorcet victory margin over the contender is only by a single vote. Maybe the Vice President can take the hit.
Here’s what else is fascinating about this: we can directly control the tradeoff between participation and Condorcet—we choose to increase the Approval contender’s margin over the B2R winner by +1 with probability P. Then the method satisfies participation under sincere ballots with probability P, and satisfies the Condorcet criterion under sincere ballots with probability 1-P.
Also, because the final head-to-head is independent, we can record the net flips in preference that favor the approval contender (+1 for flip in favor, -1 for flip in detriment). If the net is positive, we don’t need to invoke the default +1 advantage to the approval contender. That gives plausible deniability as to whether the advantage is ever actually conferred, and more closely approximates the Condorcet criterion without sacrificing participation.