You miss the point of the parent, which is that exhaling is not a chemical process. The point being that the CO2 you exhale is "somehow produced by something", and once it's there you have to exhale it.
That's a way of saying that breathing faster won't burn more fat just because of the breathing: burning fat is a bit more elaborate than that.
Out: CO2
That extra C atom comes from your metabolism. If you're in a calorie deficit, it has to come from your C atom reserve storage.