There are 6 teams, so if each team plays against the other, there will be 5 rounds for each chess variation, and totally 5 X 19 = 95 rounds. That means that the tournament will last too long time, about 2 years (if the matches will be every 14 days [=every 2nd Sat]).
Another solution would be as following. There will be 2 groups with 3 teams (e.g. A group: A.1.Murphy Chess Club, A.2.The Arizonians, A.3.Team Juanita, B group: B1.The Animal Kingdom Conservation Group., B.2The Coffee Bean Cafe and B.3.Daily Tournaments Fans Club). Each group will play against the other group a different chess variation, every 14 day [every 2nd Sat], by this way:
For the 1rst round: A.1. vs B.1, A.2. vs B.2., A.3. vs B.3.,
for the 2nd round: A.1. vs B.2, A.2. vs B.3., A.3. vs B.1. ,
for the 3rst round: A.1. vs B.3, A.2. vs B.1., A.3. vs B.1. ,
for the 4th round: A.1. vs B.1, A.2. vs B.2., A.3. vs B.3. ,
and so on.
So, there will be totally 19 rounds (equal to the 19 chess variations). Winner will be the group with more wins. The 1rst, 2nd and 3rst place will take the teams of the winner with the more wins, also.
Example of matches being teamed up for year, There are 27 Saturdays start dates.