I don't own a licence for C4D, but it should be very easy to get working if you're willing to invoke Blender on the fly from C4D. The Blender addon fully supports invoking the actions through a script too, so you can just run Blender from some C4D script to achieve import/export functionality with ease.
To auto-run a script in Blender, just invoke it this way:
%BLENDEREXEC% --python <scriptfile>.py --background -- <scriptargs>. Within such a script, call e.g.:
Then import the created FBX into whatever tool you want (all using a simple C4D plugin?).
To convert a FBX with animation to a string to the clipboard, use this script:
bpy.ops.object.rbxanims_bake() # note that this copies to clipboard, the action that generates the raw string is not currently exposed through such an api