Not sure if you are looking to animate the LightMix settings, or just need each frame to be the LightMix version and not the Beauty pass.
If the second, then you need to ensure you are saving the multipass layers and not just a single layer. That way you will end up with something like a PSD with all the layers, or end up with separate files, one for each layer (which is probably easiest for what you want). The usual C4D method for saving all Multipass layers should work just fine.