I have a suggestion... rather than stopping or limiting qualifying, just make it so that when you start a round, you either have to finish it, or withdraw from qualifying. No more starting over and over until you start -5 after 5. If you start a qualifying round, you have to play the full 18 or withdraw and be ineligible to qualify.
Another option is to not allow people to attempt to qualify if they are already under the current cut... if you make the current cut, the game denies you unless the cut drops below your current score.