The biggest problem with the issue with mercenaries is that most are just "gun-nuts" who think too highly of themselves. Consider that if you look at any issue of Soldier of Fortune magazine, most of the mercenaries, although claiming experience in Iraq and/or Bosnia, are actually a background check from being implicated in "domestic terrorism" (e.g. abortion clinic bombings, doctor shootings, et al.)
Second, for large mercenary outfits (e.g. Halliburton, Blackwater, et.al.), the problem is that they are in the pocket of certain officials (e.g. Vice-President Dick Cheney), and it would send the message that America is fighting a "dirty war". This would certainly alienate most of our allies (e.g. Great Britain, France, Israel, et al.). This would also certainly endanger the lives of any private corporation or tourists in the region. Just remember all the complaints starting in 2003 about how American troops had to protect "security forces" and mercenaries from attack by local natives..
Third, another problem with the bounty, is that those people who would be close enough to Osama bin Laden to report him in Afghanistan and Pakistan, are drug lords, who control the heroin trade. Imagine the PR/political disaster of the Washington Post and New York Times saying that the United States paid $1 billion to a drug lord who control the heroin trade in Afghanistan...