Well it would have to be considerably longer than any similar canal. Its both longer, and more elevated than the Panama Canal (Panama is 77km long, 26m max elevation, Midi is 240km long, 190m max elevation). The Volga-Don Canal in Russia, constructed in the 30s-50s come closer in scale being about 100km long and rising 88m on one side. But the locks are much smaller, about half Panamax, and its really only suitable for cargo vessels. The elevation and length of this canal combined with desired large capacity compounds the water problem. Greater elevations require more locks which require more water to feed them. So they would need massive reservoirs at the peak to maintain the canal. This could be a problem given the topography, hydrology and climate of the area between Carcassonne and Toulouse.
Really I think the issue is cost/benefit in the end. It would cost a lot to build, assuming technical and logistical hurdles are overcome its still a lot of time and money. The canal seems like it would have little commercial potential. Other than Britain bound ships coming from the Orient who would use it? And such ships would have to pay considerable tolls vs free passage around Spain via Gibraltar. So its really a strategic endeavor on the part of the French. But then their fleet would be limited to some kind of midi-max size. This could quickly become problematic in battleship construction. So much so that large ships might eventually have to forgo midi-max rendering the canal a huge albatross.
When you look at the other two examples the United States had huge resources to pour into Panama, backed by hugh economic demand and strategic imperatives while Russia had, frankly, slave labor in the form of POWs and convicts. Since there isn't really an economic imperative it seems like there would have to be some immediate strategic imperative to make it worth the effort. Perhaps a hostile Spain that could threaten passage through Gibraltar. Something that perhaps would threaten Britain too encouraging the British to subsidize the canal.