Carbon (C-12 and C-13) has about 5 barn neutron scattering cross-section, but neutron absorption cross-section in the order of millibarns. Hence, even in intense neutron flux, e.g. in nuclear reactor core, there would be very little higher carbon nuclides produced. (See chart of nuclides, e.g. https://www-nds.iaea.org/relnsd/vcharthtml/VChartHTML.html)
Even if there would be measurable amount of higher carbon nuclides: C-14 and C-15 have half-lifes of 5700 years and 2.4 seconds, respectively - neither of which is useable for the purposes you suggested.
Well, if this turns out to be feasible and usable as advertised, I would be surprised in the most positive way. But ... here is a quick back-of-the-envelope calculation:
1 gram of pure C-14 comprises of 4.3e+22 atoms. With the half-life of 5700 years, this means activity of 1.66e11 Bq. Because each C-14 decay produces 1 electron, then (assuming we collect them all) a current of 27 nA would be produced.
Obviously, one would have to dig into details but the order of magnitude is discouraging to me.
My answer doesn't involve carbon rods or beta-decay. Simply stated, any source of heat energy can be harvested to produce electricity. Having measure the gamma-ray spectra and temperatures of numerous spent fuel ponds and dry-cask storage fields; I can attest to the ample thermal energy available. Interesting idea!
Radioisotope Thermoelectric generators (RTG) have been in use for many decades for satellites (especially any that go beyond the orbit of Mars, instruments on the moon, lighthouses, navigational buoys and aids (both USA & USSR), and even cardiac pacemakers. The one i'm most familiar with use 238-Pu, half-life=87.7 years, 0.57 W/gr. The 238-Pu came from Savannah River; it was purified and tested at Los Alamos. Other isotopes like 90-Sr and 210-Po have also been used.
The isotope is in contact with a thermoelectric generator and then used to power electronic systems. The safety problems involving such materials have been well studied, in my opinion. Certainly there would be new safety and design issues to examine in using spent fuel thermal sources.
I once had two samples of 238PuO2. ½ kg an d 1.0 kg, to measure their gamma spectra and help calibrate a calorimeter. I kept them in a 55 gal drum of water that we had to carefully keep full. When I picked one out with welders' gloves, it would instantly steam as it evaporated the water. 1 kg = 577 W; it's damn hot.
Plutonium (Devil's Element) is a good source of energy RTGs need a upgrade we are relying on old techniques and technologies compared to present day. today we have superconductors graphene fullerenes carbon nanotubes yet nobody wants to make a decent research to upgrade these RTGs for Earth applications and Space Science we can dream all we want to Mars missions alongside Elon Musk during the cold times on Mars we will need in the colony an RTGs backup solution for heating and electricity.... otherwise is freezing cold down there :) during the cold nights :D