He doesn't need to spring a trap -- just sit back and let the circus go on, basking in the knowledge that it makes all of his opponents look a bit clownish.
You're ignoring the basic foundation of this controversy. It's reasonable, and totally within the people's rights to ask their president to show them his legal bona fides to hold office, when such a huge cloud of doubt resides around him.
The people aren't responsible for this controversy. Obama is. He could have nipped this entire thing in the bud when it first surfaced, if he'd simply shown the nation his original long-form birth certificate. He didn't, and he continues to refuse to do so, which puts the onus for all of this on him.
Anyone who accepts his argument that he's a legal president, while he simultaneously refuses to show this simple document, is succumbing to the perpetration of a fraud against them, and is a first-class fool.