"It's not protocol [to bring the Koreans with us into the bunker]."

My thoughts exactly. The bad guys' plan is based wholly on the assumption that the president will bring them to the bunker, which they should know if they have done their homework on the White House's security protocol. And from how well their operation goes at first, I think it's safe to say that they know the security protocol pretty well. Then how can they count on the president breaking the protocol?

It seems to be a big flaw of the movie.

I think they probably got a psychological profile of the President and figured he wouldn't risk innocent lives. Also the double agent could have told them what kind of man Asher was.

If they weren't invited into the bunker, the Koreans would have attacked the secret service right there instead of in the bunker.

They knew what kind of person the President was, and that he'd try to protect them.

Of course, politically it would be really bad for you to tell a visiting Head of State "Sorry, you're on your own" during an attack.