Error GHC-91510: illegal polymorphic type

With enough type-class synonyms

class    Objects cat x => ObjectClass cat x
instance Objects cat x => ObjectClass cat x

class    (forall x. f x => g x) => f ==> g
instance (forall x. f x => g x) => f ==> g
infix 2 ==>

class    f (g x) => (f . g) x
instance f (g x) => (f . g) x

and a few tweaks to your implementation (which I think are fixes)

type Transform ::
  (k -> k -> Type) ->
  (j -> k) ->
  (j -> k) ->
  Type
newtype Transform cat f g = Transform
  {transform :: forall x. Objects cat x => cat (f x) (g x)}

instance (Category cat) => Category (Transform cat) where
  type Objects (Transform cat) f = ObjectClass cat ==> ObjectClass cat . f
  id = Transform id
  Transform g_h . Transform f_g = Transform (g_h . f_g)

all looks to be well.