There are a few direct trains from Lyon to Carcassonne, so another option (on a working week day would be best) could be to travel in the morning to Lyon, find a cheap restaurant in Lyon near the station for lunch, then get any train after lunch either direct to Carcassonne or via a change at Narbonne.
The food in Lyon is often very good, and so should IMO be worth the extra effort and travel time.
Then supper in Carcassonne of course ...