If flying out of Dubrovnik, I'd want at least three nights there (if not flying out of there, maybe two nights, depending on what you want to do/see). That is, unless you fly out late that day from Dubrovnik. Flying days seem to be wasted days in my experience. Three nights means two full days plus arrival day (plus departure day which is probably just getting ready to depart).
A car gives you more flexibility with a short time. You could do the islands without a car, in theory. There is an early bus daily from Korcula to Dubrovnik as I recall, and there is catamaran service directly from Split to Korcula (stops in Hvar too). But I think the catamarans get in in the evening, and if you depart (say Korcula) the next morning, that gives you almost no time to explore Korcula. I can't think of any practical way to get from Hvar to Dubrovnik by public transit without taking the catamaran back to Split and then taking a bus from Split to Dubrovnik.
If one of your nights will be at/near Krka, one on an island, and three in Dubrovnik, I'd probably skip an overnight in Split. (You'd have about a five hour drive from Istria down to Krka; I guess in a long day, you could drive straight through to Krka, spend a few hours there, then drive on to Split and overnight there.) You could (with a car) stop for a few hours to explore Diocletian's Palace before say catching a car ferry to Hvar or Korcula. Or steal one of your nights from Dubrovnik. Some people love Split and highly recommend a night there. Personally, I didn't care for Split that much. Dubrovnik is extremely touristy (especially in daytime) but is also very unique.