the Federation Council ... Federation law prevented a Councillor from joining mid-session.
Would that be the yearly session?
It would be ridiculous to have a rule that required a member world to have no representation on the council for the majority of a year if their representative died (or some other event) early in a given session.
What if the member replaced their rep for some reason, a change in administration back home perhaps? They're out of the game for the rest of the year?
Even if the rule only applied to first time new member reps, it wouldn't make sense for the new member to have no voice until the next session begins. They should be a fully represented from day one.
Given the distance between stars and speed of communication and travel (passager ships might be slower than starships), it might make sense for the Federation Council to have, in addition to the member world's primary representative, a deputy/vice representative sent to the Council at the same time. Ready to step into the primary's shoes as quickly as possible.
Which ever of the authors who thought up that particular story didn't think it through.