Stuart Cove's is the cruise operator.
Stuart Cove's Cruise Ships Shark Cage Diving Bahamas Stuart Cove's Aqua Adventures - Cruise Ships Cruise port pickup times are about 1/2 way down this page:
Stuart Cove?s Scuba Diving Bahamas Nassau Dive Vacations Thrilling Shark Adventure Dive - Hotel Transfers
If it doesn't seem likely, you might go with Bahama Divers or Explore Scuba instead. Both are on/near Paradise Island, a whole lot closer than Cove's which is about a 45min. shuttle ride.
If you're off the ship/thru customs early enough you might be able to get to Bahama Divers for their 9AM dives. It's $119. A cab will be about a 10 min. ride. They're on the Nassau side of the bridge to Paradise Island. Explore is in a marina across the bridge on Paradise Island, IDK anything about them.
Bahama Divers Home Page
Home | Explore Scuba Bahamas is the only dive operator located on Paradise Island.
On Freeport UNEXSO works with cruise passengers.
Their 2 tank dive is $100. They also have the dolphin($220) and shark($110) specialty dives but IDK if those work within your schedule.
Dolphin & Scuba Diving Activities & More! They're in Port Lucaya - slightly over 5 miles from the cruise port.