Erm... to clear up some stuff....
The term 'enemy combatant' is rooted in various Geneva conventions, of which not all US are party to. (I can't remember which particular convention deals with identification of enemy combatants, non-combatants, etc.... My mind says 1947, but I'm not 100% sure).
Most of issues with detaining aliens in Gitmo bay are, AFAIK, outside of US Constitution, and mainly deals with international treaties (of which we are parties to... historically, US has refused to sign several treaties on the pretext of sovereign, and this continues to be true but to a lesser extent, unfortunately).
Furthermore, there's a saying my International Laws teacher was fond of saying: "Spies die." There is no provision whatsoever for anyone labeled a spy, and thus can be dealt with as whatever the government pleases. Shooting the feller without a trail is certainly an option. This is a reason why espionage are sometime committed by diplomatic so they can hide under the immunity (but that is not a guarantee. The only guarantee is whether the offended country prefers not to have their diplomats shot as well). As for covert ops, they are usually considered to be state-less and thus without any privileges usually given to uniformed soliders, prisoners of wars, or noncombatants.
It also helps to keep in mind that international laws aren't laws in a statutory sense as historically, a condition of statehood is sovereign, and one can hardly be sovereign if they were subject to a body of law higher than them? Rather, international laws are based on quid pro quo or conventions; if they abide by so and so, the other countries is more likely (but not guaranteed) to abide by the same. Of course, it would be folly to propose that compliance by honors is perfect (indeed, several states has been fond of pointing out others' violations while glossing over their own violations), but the ultimate restraint lies in the fact that if state wants to be given respect by other states, it will have to follow the conventions, even ones that they may not be party to.
Anyway, just some tidbits to keep in mind.