The other half move one court after each game. At the end of four rounds all those who had byes will pitch to complete their four games. All four players pitch their shoes before they move to the other end of the court. Decide before the game which end pitches first. The members are classed as in tournament, by ringer or count-all average or by a qualifying score.
All competition is within each class.
The classes can remain unchanged throughout the season, or everyone reclassed after each complete round. Six-man classes about as large as you would want. One league uses this system, plays 25-point games, and bases handicaps on the order in the standings. When I st place team plays 4th place team, the 4th place team receives 4 points. While not particularly scientific, it does even up the teams a little.