The solution is simple.
Enact a policy of “Loser Pays” and if a court case proves that Obama is not qualified then he and Michele will have to pay all the plaintiff’s legal fees.
And if the plaintiffs fail to prove that Obama is not qualified, then the plaintiffs can pay Obama’s and the government’s legal fees.
(1) It is not the responsibility of anyone but BO & the Demonrat Party to prove that he is qualified. (The Constitution does not say, "Anybody can be President as long as no one else can prove that they're not qualified.")
(2) The government should have no dog in the fight, therefore should not be a party to case, therefore should collect no expenses, regardless the outcome.
Otherwise, I think that's a fine idea.