I suppose it depends on whether the government believed the Ark had the powers detailed in the Tanakh. If they really believed it could win wars or reduce whole cities to rubble, yeah I bet they would've sent Indy that way. If they thought it was an archeological artifact, there really would've been no reason to send some college professor across the world just to antagonize a dig in the middle of the desert. The real value would be in knowing the Ark exists. The facts of its physical appearance would probably be common knowledge within months (assuming it is just an artifact and not the supernatural object shown in the movies), so again, no reason to go bother the dig site.

Also I doubt the Nazi's potential menace really mattered. Archeology rarely (read that as never) wins wars. I doubt FDR would've thought the Ark was going to really be a weapon.