The BD bypass is only for allowing the console to update with a completely dead BD drive, it doesn't enhance disc-less compatibility. It sounds to me like you have a weak or scratched/dirty laser. You can buy a new one and swap it out on the cheap, but other than that, only thing I could think if is trying the BD-mirror function in MM, although it might still give you a black screen... Either way, BD bypass isn't what you're looking for.
CECH-2001A, 250Gb Internal, 2 x 500 Gb Externals, 3.55 KMEAW CFW