You do not need to prebook ferry tickets in September. However, we always do for a few reasons - it’s one more item off our to do list, we know when we are leaving and can budget our time better, we can prepay and spread out our credit cards costs.
Renting a car for the one day would be very time consuming. You would have to taxi to the pickup point, drive, find the car drop off point, taxi to your hotel. Don’t know the cost.
Ferry port in Split is a short walk from the old part of the city. It is very convenient, especially if you are staying within the Palace walls. The arrival port in Dubrovnik is in Gruz, a short taxi ride to the old town. The tickets are fairly inexpensive.
We would rather take a relaxing boat ride than trying to drive.
We like to stay a minimum of 3 nights at any place to not feel rushed. Since you want to minimize your stops, I would suggest
Dubrovnik- 4 nights (one day to get over jet lag)
Hvar - 3 nights
Split - 4 nights
Dubrovnik- 1 night
If you do your research, you will find plenty to keep you busy in all these places. It will still be warm enough to swim also. Except for Hvar, we’ve been to Dubrovnik and Split a few times and always find new activities, places to see, and great restaurants. We took a really fun cooking class in Split.
If you want, I can recommend tour guides in Dubrovnik and Split for short walking tours or day trips.
We have never been to Hvar, we always go to Korčula. From my reading and experience, I believe Hvar has better beaches. Maybe someone else will comment about that.
I see that others are recommending visiting areas that are not easy to get to without a car, but you stated you wanted an easy trip. Adding more places with your short time frame will add stress. It’s impossible to see an entire country in just 11 days, especially one so spread out. If you enjoy your trip, another visit could be in the northern part.