This is how I got the Brass Balls Achievement in Bioshock.

This is possible even with the update.

1. With the new options, turn off the Vita-Chamber, so when you die you will be sent to Game Start Screen.

2. Save often and if you die, manually reload the last save. 

3. When you die and are sent to the Game Start Screen, remember to always Load a game (your last save) and do not select Continue 

I did this and around where the Little Sisters tackle Atlas, the achievement popped up :)