Carbon-12/-13 absorb few neutrons, since the cross-section is so low. I looked at the elemental cross-section, which is mostly C12, and it is <1e-2 b, so activation would be very little since other nuclides have much high cross sections. Even spallation reaction (n,p), (n,d), (n,α) cross-sections are low, and would also require neutron energies in the upper fission energy range or greater. The other part of the activation process is the resident time. In modern US LWRs, residence time may be about 1000 - 1100 days (two high capacity 18-mo cycles), 1320 - 1460 days (two moderate to high capacity 24-mo cycles), or possibly 3 18-mo, or 24-mo cycles, for a minority of the fuel. Many other LWRs are still on annual cycles, so some fuel can spend 5 or 6 annual cycles in a reactor, and some might go as long as 7 or 8 annual cycles.
When incorporating an additive into a fuel matrix, one has to be concerned about displacing the fuel (U, Pu, Th) atoms, particularly in an LWR.
Furthermore, in an LWR, the fuel designer has to be concerned with fuel-coolant (water/steam) chemical interactions and hydrogen pickup, more so if moves away from Zr-alloys for cladding.