Best transport from Split to Dubrovnik?

The bus (4 hours, leaving practically hourly) and the catamaran (see Krilo website) both have nice views. The bus will make a pit stop, the catamaran will have bathrooms and maybe a coffee bar. I'd go by schedule.
I really don't know what cruise people do (don't get me started), but I'd plan on getting the heck out of dodge! Maybe a kayak tour one day, if you are into that kind of thing. Maybe one day for an island tour, maybe one day to Montenegro's bay of Kotor, one day for Mostar, one day for the immediate interior.

Only the Krilo catamarans run in October and even then not through the end of the month. There's a 7:40am catamaran daily until October 20, after which it becomes a M-W-F sailing from October 21-26 and then service stops for the season. If that departure time or those dates don't work for you itinerary, then one of the frequent comfortable Pullman style buses is your only public transit option to Dubrovnik that time of year.

I took an all-day tour to Montenegro with Atlas Tours that I really enjoyed. It included Kotor, a lunch stop, Cetinje and Budva. We went the entire way around the Bay of Kotor on the way to Kotor, but took the ferry short-cut on the way back from Budva. Atlas' office is located near the Pile Gate, which was my pick-up and drop-off point. I took the tour at the end of September by which time they were already discounting and the tour wasn't offered every day, so I'm not sure how late in the season their day tours run.

My recommendation would be for the ferry, but that's because I like being on the water and find it more relaxing. I took three ferries this summer on the coast in Croatia, and they were all very easy. You do need to get in line early if you want a seat by a window. They go fast.

You can check websites as to how many cruise ships will be in town while you're there. I was lucky, the last day I was in Dubrovnik there were no cruise ships at all. Still lots of tourists, but should be better in October. Besides the day trips that RS recommends to other countries nearby, a really easy one is to take the 15-minute boat over to Lokrum Island and walk around there. Lots to see and do and the boats go back every 30 minutes, so you can stay as long or as short as you want. It was an easy way to get away from the crowds.

We took the bus and ferries around the coast depending on the time. Which ever worked better into our schedule was what we choose. Obviously, some places you can only get to by ferry. I can recommend a guide we used in Split who was excellent. He was born there and when not giving tours is a marine biologist. A very interesting person and very friendly. His English is fluent. His name is Pero Ugarkovic and his email is lignja@gmail.com. We found him on toursbylocals.com. You can read his profile there and book him through them or email him directly. Up to you.

I have heard recommendations for both ferry and bus. What are the
benefits of each?

The benefits of the bus is probably a more frequent schedule in October (and possibly multiple bus companies to choose from). I enjoyed the bus trip, the view was great and the price was very reasonable.

Frank, was your tour to Montenegro on a big bus, small bus, or van? Approximately how many people were on your tour? What was the cost?
Thanks, Karen

There are plenty of museums to visit inside the old city in Dubrovnik. Not many people seem to know this or just don’t care. Believe me, they are never crowded but very interesting. Dubrovnik is more then just walking the wall.

The Atlas tour of Montenegro was on a comfortable Pullman bus (so maybe seats for 50), but there were only about 30 people on my tour. I don't recall the exact price, but I see the tour advertised on Atlas' website for 59 euro, so about 430 kuna, and I see that it's offered on Mondays and Fridays in October 2019 (also Wednesdays in high season). Most people added a small tip for the guide and driver, which I believe was either 20 kuna each or 20 kuna for the guide and 10 kuna for the driver. Atlas had a sign out near the Pile Gate offering a discount when I took the tour in late September a few years ago. I believe the discount was around 20%, but again I don't recall exactly. I do remember thinking whatever I paid was a good value for an all-day tour of Montenegro.

The bus from Split to Dubrovnik is comfortable and inexpensive. I would recommend it.

Regarding your time in Dubrovnik ... is your lodging cancellable? If so, I would recommend staying south of Dubrovnik, in the beautiful little village of Cavtat. You can get to Dubrovnik by bus or boat to do your sightseeing, but then escape back to lovely, peaceful Cavtat and have a glass of wine by the harbor while you watch the yachts dock and the sun set. You can do tours from Cavtat -- we did the Atlas tour of Montenegro and it picked us up right in Cavtat. We also did a boat trip from Cavtat to the Elaphites islands. And then, when you're ready to come home, you're just 10 minutes from the Dubrovnik airport.

We used Day Trip service and our driver was awesome. Navigated some detours from construction that our GPS would not have done.

There is a site that you can go on to see how many cruise ships are in town on what days. Port Authority of Dubrovnik. The best advice is to go early to walk the walls, the tram, shopping etc. Then you are done when the cruise ship unloads.