I may be operating under a misconception, but can't you only be charged with treason if you are a citizen of that country? Otherwise surely he should be a prisoner of war, or if not covered by that convention an unarmed combatant.
This is espionage, not treason. Both are capital offenses, though it's not clear that it's actually legal to execute someone for these crimes in the US due to Kennedy v. Louisiana. (That ruling specifically refuses to comment on crimes against the state, but I'm sure the next person that gets a death sentence for one of these crimes will be quick to ask the court for a review.)
