You can bake animation into control rig, then edit keyframes to smoothen and make loopable.
For head animation, for example, add curves HeadRoll-HeadPitch-HeadYaw to Face_Archetype_Skeleton to animate them in sequencer. Then apply them to head bone in the post-process animation blueprint (Face_PostProcess_AnimBP) - just like LiveLink curves are processed in Face_AnimBP, but use “Get Curve Value” instead of “Evaluate Live LInk Frame” → “Get Property Value”.