Iran has numerous very quiet DE subs. Having a carrier in that little space is asking for disaster.
That and shore missile batteries would zero in on a carrier.
Station a carrier 100+ miles out in the Indian Ocean, land based assets in Kuwait, Southern Iraq, Saudi Arabia, Afghanistan to cover the Northern Gulf.
Also, I believe the Persian Gulf is too shallow for Nuke subs.
While they can submerge, they can’t deep dive for evasive action..