It's not impossible, it's just improbable. ;-)

We've had this discussion before. And my answer was to not store passwords at all. The difficult part, of course, is that this would require changing database servers - probably just as much of a beaurocratic nightmare, if not moreso, than what the OP is facing already.

This kinda reminds me of the old maxim: fast, working, cheap - pick two. Here it is: secure at the front, secure at the back, database that requires userid/password to be secure - pick two. And, like management making the first decision, management here is likely to say "All three!" as if just decreeing it changes the laws of physics.

(And here I go, revealing a bit more about my life by the assumptions that I make in my answer...)