I biked this last fall. If I’m not mistaken, both Capestang and Carcassone are on Canal du Midi, so the fastest way would be to walk the canal path. It would be an interesting walk, but the services (food & lodging) are not developed like the Chemin pilgrimage routes. It was OK by bike where I had the mobility to leave the canal for several kilometers for groceries, camping, chambre d’hotes, and gîtes, where available. It took some searching, and I ended up stealth camping a few times because there were no good choices. The services along the canal are very oriented towards the barge boaters, and a few travelers told me they were able to use the shower facilities at the barge marinas upon request, but I didn’t try it.
It’s also a stretch of the canal where the terrain is flat with few locks, and the canal winds through a lot of curves since it’s the river path rather than a dug canal. You’ll feel as you’re traveling a lot of extra distance, and I occasionally took a road shortcut to cut off a large loop.