^Actually the first episode produced in season 2 was "Catspaw." "Amok Time" was shot fifth, but was aired first because the network wanted to lead off with a Spock-centric episode.
In production order, the episodes Takei missed were #9 of the season ("The Apple"), #13-21 inclusive ("The Trouble With Tribbles," "Bread and Circuses," "Journey to Babel," "A Private Little War," "The Gamesters of Triskelion," "Obsession," "The Immunity Syndrome," "A Piece of the Action," "By Any Other Name"), and #23 ("Patterns of Force"). So I think it's likely that it was that block of nine consecutive episodes that he missed due to The Green Berets, and the other two were just episodes where he wasn't needed.
"I, Mudd" was produced immediately before "Tribbles" - George Takei does appear in the early part of the episode which was set on the Enterprise, but he doesn't appear at all once they beam down to the planet. I often wondered if perhaps he had to leave for "The Green Berets" during the shooting of this episode and thus only appeared in the Enterprise scenes (assuming those scenes were the first to be shot). I know that any given episode is not filmed in the order that it is shown, but in this case I figured that, since there are no other scenes on the Enterprise after the first act, those scenes might have been shot first.