I agree, this would be interesting. Of course it'd be best if the site code handled it, but it would be possible for someone to organize and direct one privately, and just have the players challenge each other when it's time for their game. Kind of a pain for the organizer but at least it's feasible.
Right now a Swiss tournament is going in "The Swedish Clan".
I, the clan leader and the tournament master, cannot today start the games in each round, the participants get their orders to challenge each other in an of me planned way. From next round and on, perhaps I have a solution to this.
In this tournament we have 12 participants and we run it in six rounds. In every round every player gets two games, not to heavy game load compared with the 22 games in an ordinary Berger tournament. But it will of course take some time to finish the tournament.