A friend of mine was a hobbyist blacksmith for many years (and ran a blacksmith shop at the local Renaissance Faire); I did some smithing with him from time to time. The above feels like the right answer to me. Compared to a coal-burning power plant, or even a diesel truck, I suspect that your impact is small, but non-zero.
My friend eventually built himself a gas-fired forge, which ran on propane, IIRC. If you’re really concerned about your carbon footprint, I suspect that propane might be a cleaner fuel source, though it’d require building a new forge.