When you King's Court a King's Court, when must you pick the three target cards?
Short answer: You do the "one at a time" option as you described.

Long answer: 

This case is precisely identical to the more common question of using Throne Room on a Throne Room, as the two cards are worded identically except for the number and a "you may".

The official Dominion rules PDF says:


  You completely resolve playing the Action the first time before playing it
  the second time. If you Throne Room a Throne Room, you play an Action,
  doing it twice, and then play another Action and do it twice; you do not
  resolve an Action four times.


In your case, this means "completely resolve playing King's Court the first time before [starting to] play it the second or third time". This means that yes, if your first Courted action was a Smithy, you're going to have lots of choices for the second and third King's Court.