HOWEVER, we know that only 31% of the decay energy is actually carried away by the beta particles, and the rest ends up as heat (which is not captured)[1]. On top of that, the efficiency of converting the beta particle's kinetic energy into electrical power is low. I found a paper citing a 26% conversion efficiency [2], which seems high but lets go with it. Finally, the mass of the battery obviously includes more than just the C-14. Lets assume that half the mass of the battery is C-14.
That would bring us down to 300 microwatts per gram of C-14, and less than 0.05% of the specific power of a LiIon battery.
[1] There's also a positron that shares half of this energy -- I don't know if these devices capture that, so I'm just leaving it in.
[2] https://www.sciencedirect.com/science/article/pii/S016890022...)