Assuming the exterior of the mesh is being cut during the fragmenting process, its UVs would be cut and preserved as well, allowing whatever texture you initially made to carry through. It’s possible that the UV data is lost during the fragmenting, in which case you could possibly transfer UVs from an unbroken mesh to the finished (hopefully this isn’t the case). The interior surfaces would receive new UVs (if any). You can probably auto-UV those pieces and assign a separate material to them.
I haven’t worked with C4D but that workflow checks out.