The firepit would have to be reworked to have a stoked vs. non-stoked condition. Fires will burn to embers and then extinguish unless you push new fuel onto the embers. Make the fire unable to cook if not kept up to temp and burn at the slower rate, then to cook, poke it with a stick/firepoker which is boost both temp and fuel consumption for a short length of time. Once that time is done, the fire returns to the lower rate of consumption and the heat subsides.
You could match the higher consumption to the the number of pieces of fuel used now to cook. If a pot cooks using 5 units of fuel, poke it 5 times to burn hot that long.
If nothing is cooking, a poke should return/bank the fire to the lower rate.