If memory serves, some of the original WTC bombers were tried and convicted domestically -- as far as I know they weren't citizens. If that's right, there is at least precedent to hang this on, even if it doesn't clarify your questions re: justification or process.
Just speculating, I'd say trying them in US courts does two things; 1) lessens the likelihood that a terrorist will be released back to his native land and suffer fewer -- or no -- consequences for his crime there, and; 2) serve as a symbolic band-aid (covering the last administration's terrorist-recruiter's wet-dream of a policy) showing that our government is superior and more just than the one Jihadists propose. Regarding #2, I can't help but feel it's too little, too late.
Also regarding #2, I know people like Whott will complain that this represents some form of appeasement. I would disagree with that position because we aren't talking about strategy between two sovereign nations, in which concessions might be taken as signs of weakness. Instead we're talking about making it harder for extremists to recruit new members -- we aren't, after all, going to change any Jihadists minds with our policies, but if we can give them less to work with as far as comporting ourselves within the letter of our laws and ideals is concerned, the movement stops being able to hang on our hypocrisy and dies like any other death cult.
This "war" can't be won on a battlefield when the enemy hides among the innocent -- the only way we eliminate the threat is to win hearts and minds (along with some surgical wet-work, if need be).