We could let him rot in a cell, but that would mean a trial, which he would probably just use as a platform to run off at the mouth, as Saddam and Milosevic did. While I would like to see him treated as the criminal that he is, I don't think a trial would be good for anybody.
A public execution, or even a public announcement that we had killed him in an airstrike or covert operation would just make him a martyr.
I think, if we did find him, and this should be a priority, we should kill him on the spot, cremate the body, and keep the whole thing as secret as possible. Though killing him would be great news for the US and our friends around the world, keeping it a secret would deny him martyr status, we wouldn't have to watch a long drawn out circus trial, and there would be no legitamate successor.
How could anyone try to take his place in the public eye if the public didn't know that he was dead? His closest followers would probably realize that he was dead, and try to make an announcement, and name someone to succeed him, but we could spread a rumor that there was an internal power struggle, and one faction tried to off him, and he was in hiding trying to recover from his wounds.