The Mistrals were built to a commercial standard, which affects things like internal ventilation (how well the ship can be sealed in an NBC environment), construction of water tight doors, the ability of lighting and electrical fixtures to withstand shock from explosions in other parts of the ship or in the water close aboard, fire safety features, fire fighting equipment, counter flooding capabilities, the presence or absence of high volume pumps to de-water flooded compartments, etc.
I had not read anything to indicate the Cavour or the RN's CVF are built to a commercial standard (and, btw, there are a dozen or more commercial standards to choose from, each with it's own associated detailed specifications, insurance rates for the type of commerce your ship is engaged in, and operators make an economic decision about which standard they wish their ships to be built to). The USN's first purpose built LPH's of the Guadalcanal class were built on a commercial cargo ship hull and machinery but were otherwise mil spec. Note that Tripoli struck a mine in the Persian Gulf during Desert Storm, suffered a huge hole in the hull some 7 X 10 meters, the shock of which blew the fire out in the boilers. After isolating the flooded compartments and re-lighting the boilers, the ship continued on it's mission. This wasn't even a double hulled ship.
Most anti ship missiles will only damage a large ship like a modern tanker or container cargo ship. The old oiler USS Ashtabula was a WWII vintage AO that was lengthened, or "jumboized" to 25,450 tons. This is about the size of a Mistral. After decommissioning Ashtabula was used as a target in a SINKEX. After being hit by 8 Harpoons, 2 Standards, 3 Sea Skuas, 4 1000 lb bombs and over 100 rounds of gunfire the ship was ravaged but still afloat. Have a look at the photo.
The Sacramento, which I put some time aboard, had double bottoms, and the fuel tanks ran along the sides of the ship. Ammo and dry stores were down the center third of the ship, surrounded by fuel tanks and void spaces. The fuel tanks had longitudinal baffles to reduce slosh, and also to take the impact of weapons. Each tank had automated fire fighting systems. The cargo holds ended two decks below the weather decks, with cargo handling decks above the holds, so they were protected from bomb and missile damage. Elevators served the cargo holds and brought stores up to the cargo handling decks.
These were very well protected ships and their crews were very confident of them. They were fast ships too, able to outrun the frigates of the day and even some destroyer classes.