I wouldn't use charcoal as medium. If for no other reason, as it dissolves in the water would turn it black and leave residue that is likely to clog the
pump. But I'm sure there are other drawbacks as well, like
nutrients bonding with the excess carbon and becoming unavailable to the plants. I also wouldn't use the type of charcoal for filters (it's a completely different type of charcoal), this charcoal is specifically used to take elements out of the water, that would defeat the purpose of the
nutrient solution.
But you could use regular wood chips (no charcoal included) even sawdust, as long as it's untreated. You just don't want it treated with herbicides, pesticides or fungicides. You also don't want any other chemicals in it like paint chips. Lastly, if unsure I would soak some in water, then test the pH every day for a week or so, just to be sure it was pH stable.