You are looking at this the wrong way.
Answers posted before the bounty is started aren't eligible for the auto-awarding processes, but there's nothing stopping the bounty being awarded manually. If you delete your original answer and repost the same answer a) there's no guarantee you'll get the 2+ up-votes necessary to be eligible for the auto-award process and b) there's every chance that 10K+ users and moderators will spot the behaviour.
If your original answer hasn't helped the OP (despite being up-voted by others) you can either a) delete it (as you originally suggested) and get the Disciplined badge, b) edit it to include the new answer (but perhaps leave the old information there as a reference), or c) add a new answer.
If your answer has 4 up-votes then there's a good chance that the the bounty poster will award your answer the bounty anyway.