I have seen plenty of 'experiments' online (e.g., https://paksc.org/pk/diy-projects/build-your-own-battery-cell-from-activated-carbon/).
I could not find a proper Life Cycle Assessment but based on the Technics people use it looks the homemade charcoal battery has larger environmental footprint than the commercial one.
It is fine that you consider charcoal as close to carbon neutral (it is not), but there are efficiency, yield and larger waste management issues of the homemade battery. CO2 is only one effect among the plenty environmental ones.