I believe the navy kept the CVs out of the Gulf during Preying Mantis, and simply launched their aircraft. The Gulf of Sidra was a different story as it involved a challenge over the definition of international waters and the Gulf itself was merely an extension of the Med and was not, as the Persian Gulf is, a confined body of water, only narrowly connected to the Arabian Sea.
Praying Mantis was not the only event that took place in the Persian Gulf during the 1980s. You've apparently forgotten about Earnest Will and Nimble Archer not to mention all the deployments into the gulf when offensive operations did not occur. In addition MEUs deployed to the Gulf as did the French Carrier Clemenceau during the Tanker War. American carriers operated in the Persian Gulf long before Desert Shield and Desert Storm.
USS Ranger CBG underway in the Persian Gulf during Operation Nimble Archer, December 1987.