A coppiced tree also rapidly regrows it's removed biomass to regain that balance so I wonder how much available material would be left for anything else. My guess is that the tree would reabsorb whatever was lost to the soil and that removed biomass (from the coppicing) would be lost to the system. Unless it is added back, of course, in some form. That's where being a nitrogen fixer might help. It comes from the air with the carbon dioxide so there is gain to the system there. I'm just thinking through my keyboard. My answer is I don't know but it's interesting to consider. My produced biochar gets mixed with compost, rabbit manure and other inputs and added back to the soil.