I don't think there are any fair schedules for your scenario. The way I read it your tournament is played on one day only, so the number of rounds will be sufficiently limited that no balanced schedule is possible. If you think of the tournament from the perspective of one kid, then each round they play in, they will have 3 partners on their team, and 4 opponents on the opposite team, the fact that there are more opponents than partners, excludes the possibility of having each player once as both partner and opponent. There are some schedules that will work but in general they will have as many rounds as there are players. The only one I have at the moment is the one below for 17 players and 17 rounds with one bye per round. Here players have all possible partners exactly 3 times, and all opponents exactly 4 times.
(14 17 5 2 v 10 16 9 3) (13 6 15 4 v 7 8 11 12) ( 1)
(15 1 6 3 v 11 17 10 4) (14 7 16 5 v 8 9 12 13) ( 2)
(16 2 7 4 v 12 1 11 5) (15 8 17 6 v 9 10 13 14) ( 3)
(17 3 8 5 v 13 2 12 6) (16 9 1 7 v 10 11 14 15) ( 4)
( 1 4 9 6 v 14 3 13 7) (17 10 2 8 v 11 12 15 16) ( 5)
( 2 5 10 7 v 15 4 14 8) ( 1 11 3 9 v 12 13 16 17) ( 6)
( 3 6 11 8 v 16 5 15 9) ( 2 12 4 10 v 13 14 17 1) ( 7)
( 4 7 12 9 v 17 6 16 10) ( 3 13 5 11 v 14 15 1 2) ( 8)
( 5 8 13 10 v 1 7 17 11) ( 4 14 6 12 v 15 16 2 3) ( 9)
( 6 9 14 11 v 2 8 1 12) ( 5 15 7 13 v 16 17 3 4) (10)
( 7 10 15 12 v 3 9 2 13) ( 6 16 8 14 v 17 1 4 5) (11)
( 8 11 16 13 v 4 10 3 14) ( 7 17 9 15 v 1 2 5 6) (12)
( 9 12 17 14 v 5 11 4 15) ( 8 1 10 16 v 2 3 6 7) (13)
(10 13 1 15 v 6 12 5 16) ( 9 2 11 17 v 3 4 7 8) (14)
(11 14 2 16 v 7 13 6 17) (10 3 12 1 v 4 5 8 9) (15)
(12 15 3 17 v 8 14 7 1) (11 4 13 2 v 5 6 9 10) (16)
(13 16 4 1 v 9 15 8 2) (12 5 14 3 v 6 7 10 11) (17)
There is no simple formula that will solve all these scheduling problems - it is more a case of building solutions for specific problems.
It's not a problem to balance the 16 players so that they play with each other on the same team exactly once - the issue is that you can't also balance the opposition. For example this is about the best that can be done:
(15 10 13 3 v 1 2 16 11) (12 8 6 5 v 14 7 9 4)
( 2 8 10 14 v 4 5 13 16) (11 12 7 3 v 1 9 15 6)
(12 13 1 14 v 4 6 11 10) ( 2 5 3 9 v 16 15 7 8)
(14 15 5 11 v 6 13 7 2) ( 8 1 3 4 v 10 16 9 12)
(12 4 2 15 v 6 14 3 16) ( 5 10 7 1 v 11 8 9 13)
within teams the balance is perfect, however any player has 3 other players who they never oppose, and 2 other players who they oppose 3 times. For example player 1 never opposes 2, 5 or 14, but opposes 10 & 11 three times. So the kid who never opposes one of the best players and opposes a weaker player three times, has an unfair advantage.