Does anyone have recommendations on the best way via car from Dubrovnik to Hvar?
One option is car ferry, Split to Hvar, then Hvar to maybe Korcula, then drive to Dubrovnik. The other option would be to leave the car parked in Split, head out to Hvar on passenger ferry, then back to Split and drive to Dubrovnik. Either way, you will have some costs (parking or ferry or both) in addition to rental costs.
The other thing to check out is if the rental agency you use has any issues with taking a car on a ferry, it is a common restriction to not allow ferries.
To drive from Dubrovnik to Hvar, there are two obvious options.
Follow the coast road to Drvenik and take the ferry to Sućuraj on the eastern end of Hvar. Drive along Hvar to your destination – Hvar town, Stari Grad or wherever. Some visitors feel that this road is too narrow. I have driven the route five or six times, and while it is narrow in parts, if taken slowly, it is perfectly safe.
Take the Split – Stari Grad (Hvar island) ferry. To get to Split either follow the coastal road all the way, or use the A1 motorway which starts near Ploče.
Ferry details can be found at http://www.jadrolinija.hr/en/sailing-schedule/local-lines-30-05---28-09-2014 .
By all means check with the car hire firm, but in my experience Croatian firms do not have a problem with cars being taken on Jadrolinija ferries.
Sounds like you've received some good advice. Happy driving!