Are you also saying that this cannot be resolved by some kind of a
type families-based type-level 'flip'?
It really can't, no. You can define
type family FlipF f a b where
FlipF f a b = f b a
but FlipF, being a type family, is not first-class. You can't make any instances whatsoever for FlipF f a -- you'll probably get an error about a partially applied type family. What you *can* do is make a Flip newtype:
newtype Flip f a b = Flip {unflip :: f b a}
This behaves perfectly:
instance Bifunctor f => Functor (Flip f a) where
fmap f = Flip . first f . unflip
instance Profunctor f => Contravariant (Flip f a) where