Let’s try it and see what happens. I am doing this on the fly, without any idea where it will end.
08:30 uber to Sitx Rental Car agency (30 min, wait and transfer) and rent a car (30 min for paperwork)
09:30 Depart for Sixt Rental in Split to return the car. (3:15 drive if direct with no stops and no traffic issues or making wrong turns and getting lost)
12:45 Earliest you will reach Split Sixt Rental. Return car (30 min) and take uber to ferry port (30 min wait and transfer).
The earliest you could be at the ferry is 13:45
You said use time till midafternoon for sightseeing along the drive. I will say that midafternoon is 15:00. So, you have 1:15 for sightseeing and eating lunch between Dubrovnik and Split.
I am not saying this is the best source for ferry schedules, but good enough to have an idea that they are frequent. Whatever time you get to the port, you can be on a ferry in 30 minutes. https://www.ferryhopper.com/booking/results?itinerary=SPA,HVA00&dates=20260917
Now, if you hired a driver, you miss the freedom of driving the road. That’s a really big plus for renting a car. If that is the driving force, then it trumps everything else and nothing else matters. Its your holiday, it doent have to be anything differnet than what you want it to be.
Sixt wants 300€ to 365€ depending on how much insurance you want to buy. With a couple of uber transfers you will be at 400€. Yikes! Croatia has gotten so freeking expensive. There are probably cheaper though. But of course, if money were the issue, you would be on that 10€ bus.
So, what does that private transfer with some stops cost? Worst case scenario is 400€ with 2 stops of an hour each; one at a winery and one at Ston (there are other options for stops and other providers - I've had a couple of different drivers in Croatia, all worked for me).
But driving yourself will be more efficient … except that the transfer driver knows the roads and with the transfer driver its door to door so no ubers to rental offices and no paperwork and no financial risk if the car is in an accident. So you could be picked up at the gate to Dubrovnik Old Town at 8:30, make the drive including THREE + hours of sightseeing and be at the ferry dock at 15:00. Maybe meet a nice Croatian chap to boot.
Oh, you asked about places other than Ston? Ask the driver, I suspect he might know some stuff thats not in the guide books. But there are some wineries and i seem to remember an interesting olive plantation.