There should be no limit to the number of people in a clan. It is really the major social aspect of what is a fairly simple single player game. You should even be able to join more than one clan. Make it easy for communities to develop in game.

I think moving to a set of clan vs individual challenges would be best. Make people be forced to choose how they want to allocate their actions- for group benefit or personal? Some people will want to be the best in their bracket, but that will come at a price of hurting clan performance. Competitive vs non competitive clans will evolve out of that, and game mechanics won’t need to be introduced to handle it.

Don’t make people make an explicit choice every week/month/leap year whether or not to participate in aspects of the game. If people notice that their clan’s bracket is especially weak, then they should be able to take advantage of that, without having to wait for the next round (when the weakness may not exist) or for the clan leader to push a bumon or etc.