I am trying to create a mouth unit, which can be attached to different objects, and make the mouth speak with lip sync.

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Why not do the opposite...
Extract the mouth with blend-shapes from CC character and return in back to CC as a Humanoid.

This one took me 10 min to make: https://drive.google.com/file/d/1dpUvOPxsNbRPpJrFQlYfqoWz8mx9UMD_/view?usp=sharing
(it has Kevin's talking animation)

You basically export a CC3+ character with Mesh and Expression Sequence.
Import to Blender and delete all mesh except the desired shape of the mouth (I left teeth and tongue as well). Mouth could be reshaped as well.
At least 2 faces for the feet bottom should not be deleted (one for each feet). This is for character stability. Delete unnecessary materials.
Import this character back to CC, switch to Extended face profile and then import blend-shapes with "FBX with Frame sequence" from the same file you import the character with.

This is an interesting direction. thank you very much.
I'm aiming for a toon mouth (see reference image), and I'm wondering how much can the realistic mouth be changed before it breaks.


I assumed that it would be possible to define the shapes I created, as expressions on top of an existing skeleton. It is not possible?

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(Lao Tzu) 

Something like this could be made with just mouth/lips morph sliders in CC.



Then in Blender it could be further tweaked, subdivided and teeth extruded from the lip mesh up and down.
It should work fine.

Making a character from an existing mouth/parts is probably possible, by weighting to CC3 armature in Blender (particularly to Head bone) and importing to CC as Humanoid.
Then importing Blendshapes separately from the same file as Custom and then reassigning them one by one to Visemes.

Embrace simplicity

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(Lao Tzu) 

It was a loooong day )

Embrace simplicity

Reduce selfishness

Have few desires

(Lao Tzu)