2 years ago I was on an FLL team, and last year I mentored 2 FLL teams, so here is my advice.
Team structure is crucial to the success of an FLL team. Most of the time, all 10 kids want to work on the robot. It is normally a good idea to section off parts of each meeting to do the research project.
Make it very clear to the kids that the entire table will not be completed. Make them set goals and plan ahead of time. Try to encourage them to come up with a game strategy (Do we go here first, or should we maybe go there?). Most of the time, this should be done by them.
This video is probably a good starting point for that.
During a break (snack time is perfect for this), run core value sessions. Create group challenges for the them to do. I even took a rubric and marked them on how well they did, similar to the real core values judging session. Remember its not about the challenge, but how them came up with the solution.
1 week before the competition, try and enforce a stop work day. After this point, only practice and final tweaking should be done. Any mission or project idea added now will only make everything else suffer. During practice of the table missions, play loud music and have the other kids yelling and cheering. Try and replicate the conditions of the tournament venue (and they can get insanely loud).
At the competition, make sure you use the practise time at the table wisely. Make sure light sensors are calibrated, and all the missions are functioning. Remember that there will be subtle differences from the table mat to your mat at home. Preach to the kids that they are here to have fun. Talk to other teams, learn from them, and teach them too.
I'm sure I'll come back to this thread, because there is no way I can sum up all my advice in a few hundred words.
P.S. If you really want to tip the teacher-mentors over the edge, buy him/her some
Core Value Shoelaces.