First time in Croatia. 9 days during the first week of October.

We used Octopus Transfers in Croatia three times and would recommend them. https://www.octopustransferscroatia.com

Note that Plitvice is a fair distance from Split. Another option much closer to Split is Krka.

The bus is probably the easiest way to get between Split and Dubrovnik if you are going point to point - there are several a day. See the BusCroatia website. The main bus station in Dubrovnik is not by the old town - you will need to take a local bus up there as I recall (I didn't actually take the bus between Split and Dubrovnik - I took a bus from Dubrovnik to Korcula.) Of course, you can also take the catamaran back if the schedule works out.

Do you already have plane tickets booked? No way to fly into Split out of Dubrovnik or vice versa? That would save you a lot of ground travel time, though if it costs 2X more or adds an extra flight on one end, probably not worth it.

If you really do want to see Plitvice, renting a car in Dubrovnik as you are heading north (instead of taking a bus) and driving up to Plitvice, spending a night, seeing the park in the morning before the crowds arrive, and then dropping the car in Split makes some sense. Driving in Croatia is easy, outside of the cities. However, you might consider being flexible with your reservations based on the weather. It starts to turn colder and more rainy in October, but of course it varies every year. A few years ago, I got caught in some violent rain storms in Korcula and Split in the first week of October. I went back last October in the first week, and I didn't have a drop of rain - just mild temperatures and sun. (But it had rained heavily a few days prior to my arrival.) In any case, I wouldn't want to trudge through the mud and over rainy boardwalks in Plitvice in the rain, though it's a gorgeous park. There is another park closer to Split, Krka National Park, which is not really the same, but there is one huge waterfall there, Sradinski Buk, that is bigger than anything at Plitvice (though I greatly prefer the overall experience at Plitvice, which is really something.)

On the way back to Split you could stop in Zadar or Sibenik, a lovely town that's less touristy than Zadar.

Everything will be open in Hvar in the first week of October, don't worry, just with fewer crowds. That's the only time I have ever been there (last October). There were still plenty of tourists (and weather was nice), but I'm guessing it's a mob scene in July and August by comparison.

The local bus schedule on Hvar had just changed to the off season schedule the day I arrived to fewer buses to Stari Grad, and the bus schedule the TI gave me was wrong (so I missed the bus to Stari Grad and had to wait hours longer than planned).

Hvar Town was beautiful - I did only a day trip there. Supetar, Brac was very pleasant and nice but nothing that memorable. (I took a bus to Milna for a few hours - quiet little town, kind of a shrug.) However, the island that really blew me away was Vis, enough that I went back after a day trip and spent a few nights there. It seemed somehow less busy and not as touristy (though it was off season). Vis just enchanted me for some reason in a way the other islands didn't.

Agree, I would try to visit Krka instead of Plitvice.
You could maximize your time a bit by departing on arrival to one of your other destinations, like Hvar. Of course, catamaran schedules may dictate.

Take a catamaran from Dubrovnik to Split. There are 2x per day. If you take Krilo it is 4.20 hours. Check croatiaferries.Com