Which chemical reaction to use to remove carbon/peat from sieved earth/gravel?
It's easier to use flotation or froth flotation to separate out the less-dense peat (silt?) and carbon (I assume from the activated-carbon filter) than to use a chemical reaction..

Simple flotation would depend on making a dense solution (e.g. saturated Epsom salt, $\ce{MgSO4}$) and shaking the gravel mix in it... if the liquid's density is greater than the average density of silt with its entrained air, the gravel will sink and silt float, to be skimmed off. You can reuse the liquid for more separation.

Froth flotation relies on a surfactant wetting the silt and keeping it suspended while the gravel settles out. Try a bit of hand dish-washing
detergent.