Yea, I think you're mostly right.He's the President of Nothing. The United States doesn't exist any more. He has about as much authority as the President of the USSR did in 1993.
Well technically he'd have authority over any remaining US armed forces. While regional governments after an alien invasion may not recognize the authority of a national government it would be interesting in such a scenario how individual combat units would respond to announcement that the President is alive and in control.
I've never been in the military but as I understand it they are indoctrinated to follow the chain of command and if commanders had allegiance to the President - so would they - IMO?
As Guy pointed out, if the term he was elected for is still active, without a question, he is the President and it is his duty to lead.
If the term he was elected for has expired, during the invasion/rebellion, then he is the natural one to lead, and I believe he has a duty to do so, however, if there is a mechanism for objecting and replacing him, then it becomes stickier.
In addition to all that, The Federal Government is not the ruler of the United States, the States are self Governing, and merely have to stay within the confines of Federal Laws. Other than Federal laws, the only "authority" over the states comes from the power of the Military and the power of the purse. Well, obviously, the power of the purse is gone, and there is no infrastructure for enforcing Federal Laws Countrywide, so, really, local leaders aren't practically bound by the President's authority (Other than his possible ability to take his authority with the military). Of course if a President does turn up, it would be in everyone's best interest for the Local Leaders to work with the President and allow him to take the lead coordinating with the various local groups (providing he isn't some "off the rails" fool)