Unfortunately no balanced doubles schedule for 6 players can exist - mathematically it is impossible. There will always be asymmetries, for example in this schedule:
[(B E):(C D)]
[(D F):(A B)]
[(D E):(C F)]
[(B F):(A C)]
[(A F):(C E)]
[(B C):(A E)]
[(B D):(E F)]
AD never partner and EC oppose three times. The other pairs of players partner once and oppose either once or twice.