One of the "issues" that Nassau has is that it is a stop for just about every cruise ship in the Caribbean it seems. Because of this, the two main dive ops are hamstrung. They absolutely must get the divers from the cruise ships back to the boat before the boat leaves Nassau. The impact that this has on shore based divers who are there for several days is that their schedules are now being constrained by the limitations imposed by the cruise ships. Read that as artificially short dives to hit a known, fixed timeline. This timeline starts when the Dive Op picks the customers up at the docks and it ends when they drop them off at the docks at the end of the day. Everything else (filling out paperwork, gear rental if req'd and returning gear at the end of the dive, the drive from the cruise docks to the dive op and back at the end of the day, etc.) has to fit inside of that window and there is no room for error. The cruise ships will leave when they are scheduled to whether the customers are back on board or not, and then it becomes a matter of who pays to get them to the next stop to meet back up with their ship. There are potentially huge financial consequences if they are not back on time.
There are a couple ways that you might be able to mitigate this. If you can join up with a group that will be there at the same time as you that has enough people that they have their own dive boat, then there may be more flexibility in the schedule. (I was a part of a group that was using Stuart Cove once and we had our own boat as well as the same Captain and Dive Master all week. We were able to pretty much set our own schedule within reason.) Keep in mind that we were a big enough group that it was viable to supply us with our own ground transportation to/from our hotel. A big group buys a lot of flexibility and if you can get "adopted" by one for the time you are in Nassau, it might serve your needs.
Another way to extend your dive time is to play the edges so to speak. I wouldn't try this until they get to know you a little, but I have been able to gear up and be ready to splash as soon as the boat stops. You might miss the dive site briefing, but you might get an extra few minutes underwater. DO NOT try to extend your time at the end of the dive, like I said, they are on a hard schedule that is dictated by the cruise lines and they have little to no leeway in hitting those timelines.
@ratjn you start of by saying "we are ...". How many people are in your group? If there are enough, then it might be worth it to contact the two major dive ops now and see what can be worked out for a group your size or if there are any groups that you could join up with that will be there at the same time and would alleviate the cruise ship timeline.
Communication is critical. I can not stress that enough. Talk to the two main dive ops. The bulk of their business is based on the cruise ship divers and that is what they tend to gear their business model to accommodate. Maybe, just maybe, there might be a way to make it work more the way you are hoping for or at least some middle ground.