I made a bit of conceptual progress with a potential proof (or route to a counterexample?) of participation for approval vs. B2R.
The setup is as follows: E is an electorate, C is a candidate set, and v is some single new voter that when combined with E forms E'=EU{v}.
The electoral processes of interest are then E(C) and E'(C), where we hope that participation is satisfied in the sense that if the winner of E(C) is W, and the winner of E'(C) is W', then v does not strictly prefer W to W'.
We can consider the sequence of B2R losers L1, L2, ... Lk, and the corresponding L1', L2', ... Lk'.
Then since under B2R the first k losers can never include the top-sorted candidate who is the adversary of the B2R survivor, it follows that the participation property will propagate by an inductive hypothesis if at any point (as in for some k), the sets {L1, L2, .. Lk} and {L1', L2', ... Lk'} coincide.
It could also be possibly useful to know that B2R is equivalent to BNR. Furthermore, the methodology is only really of interest when there are more than two candidates.
In any case, I think this method makes sure that the winner is either a highly approved candidate, or a candidate that a majority prefers over a highly approved candidate, which to me is an interesting guarantee. I still have not been able to find an example of participation failure... I have found instances though where the Condorcet loser is elected. Still, the Condorcet loser criterion seems only possible to fail by small chance when small electorates vote over very few candidates.