The problem with BVP in my opinion is that you will rarely find a large enough sample size to justify your argument that it exists, but I can’t all but write it off as something that doesn’t happen.
Can a pitcher/catcher battery, with good knowledge of the batter they are facing, have an edge over that batter depending on the pitch thrown? I’m basing this on zero research, apart from watching the majority of at-bats during the MLB season (I take the All Star Break off), but it seems like knowledge of a batters pitch preference combined with a pitcher that can exploit said preference would lead to a 1v1 edge.
Not exactly BvP but at the same time, if I see a pitcher that consistently owns a batter, I’d be likely to shy away from that batter provided he’d face the pitcher more than a handful of times. Further, believing that a psychological advantage or disadvantage doesn’t exist between specific players is shaky ground to stand on.