Hello!
Do I have to use my real name on my FB and other social media accounts to participate in a bounty program?
You definitely not obliged to use your real identity. I've never heard of any major bounty campaign asking people to use only their real accounts. That would be ridiculous, since it is legal to use alias or a nickname for twitter, FB and any other social network.
Yet you may benefit from your social networks accounts, if they have a lot of friends, subscribers, followers etc. Many bounty campaigns pays "influencers" way more, than ordinary users.