No matter how you set this up there is bound to be some unfairness. My preference would be for giving everyone the same number of games, so I would choose to play 20 rounds where each player gets exactly 16 games and 4 byes. This can be done so that all pairs of players partner at least once and at most twice (so two repeated partnerships per player), also that all pairs oppose at least twice and at most three times.
(15 12 v 6 1) (10 11 v 3 5) (13 4 v 7 14) ( 2 8 9)
(11 13 v 7 2) ( 6 12 v 4 1) (14 5 v 8 15) ( 3 9 10)
(12 14 v 8 3) ( 7 13 v 5 2) (15 1 v 9 11) ( 4 10 6)
(13 15 v 9 4) ( 8 14 v 1 3) (11 2 v 10 12) ( 5 6 7)
(14 11 v 10 5) ( 9 15 v 2 4) (12 3 v 6 13) ( 1 7 8)
( 3 1 v 7 5) (14 10 v 9 12) (11 6 v 8 2) ( 4 13 15)
( 4 2 v 8 1) (15 6 v 10 13) (12 7 v 9 3) ( 5 14 11)
( 5 3 v 9 2) (11 7 v 6 14) (13 8 v 10 4) ( 1 15 12)
( 1 4 v 10 3) (12 8 v 7 15) (14 9 v 6 5) ( 2 11 13)
( 2 5 v 6 4) (13 9 v 8 11) (15 10 v 7 1) ( 3 12 14)
( 9 1 v 12 11) ( 5 4 v 6 8) ( 7 3 v 15 2) (14 10 13)
(10 2 v 13 12) ( 1 5 v 7 9) ( 8 4 v 11 3) (15 6 14)
( 6 3 v 14 13) ( 2 1 v 8 10) ( 9 5 v 12 4) (11 7 15)
( 7 4 v 15 14) ( 3 2 v 9 6) (10 1 v 13 5) (12 8 11)
( 8 5 v 11 15) ( 4 3 v 10 7) ( 6 2 v 14 1) (13 9 12)
(15 9 v 10 6) ( 7 11 v 14 4) (12 5 v 2 13) ( 1 3 8)
(11 10 v 6 7) ( 8 12 v 15 5) (13 1 v 3 14) ( 2 4 9)
(12 6 v 7 8) ( 9 13 v 11 1) (14 2 v 4 15) ( 3 5 10)
(13 7 v 8 9) (10 14 v 12 2) (15 3 v 5 11) ( 4 1 6)
(14 8 v 9 10) ( 6 15 v 13 3) (11 4 v 1 12) ( 5 2 7)