Literature reference for search-BPP
I’m not sure about the exact definition as given. However, the kind of search problems that has been studied the most in the literature are NP-search problems. In this context, there is no meaningful difference between “BPP-like”, “RP-like”, or “ZPP-like” randomized polynomial-time algorithms, as we can check the correctness of any purported solution in deterministic polynomial time. Thus, while the class of NP-search problems solvable in probabilistic polynomial time has been studied for a long time, it has not been generally called “Search-BPP”. In particular, the class is mentioned in Papadimitriou’s seminal papers [1,2], where it is denoted FZPP.

[1] Christos Papadimitriou, On inefficient proofs of existence and complexity classes, Annals of Discrete Mathematics 51 (1992), pp. 245–250, doi: 10.1016/S0167-5060(08)70637-X.

[2] Christos Papadimitriou, On the complexity of the parity argument and other inefficient proofs of existence, Journal of Computer and System Sciences 48 (1994), no. 3, pp. 498–532, doi: 10.1016/S0022-0000(05)80063-7.