A bounty? No. Just give the money as grants to academic research labs applying to do a search for new antibiotics.
Bounties are quite effective compared to long term grants just because of human nature. Which gets quicker results:
"Hello, research foundation, we'd like to give you a nice annual salary as long as you present annual reports on how well you're progressing"
Or: "Here's a whopping pile of cash that's all yours as soon as you come up with the goods."