7 Nights split between Dubrovnik and Split

Do you have 7 nights or 6 nights in Croatia? If you have 7 nights, I'd probably spend 3 nights in Dubrovnik, 2 nights on an island like Hvar or Korcula (or Vis - a little harder to get to), and two nights in Split. If you have only six nights, maybe I'd split them evenly between Dubrovnik and Split and do day trips by catamaran to islands e.g. to Hvar. No need to rent a car at all - you can get between Dubrovnik, Korcula, Hvar, and Split by catamarans.

Keep in mind that you will be visiting at the absolute peak season, so expect big crowds in all of these places.

We arrived in Split by ferry from Italy, spent a couple of days, then took another ferry to Hvar for a couple of days, back to split by ferry, then bus to Dubrovnik. I would advise checking cruise ship schedules on one of the various websites to see what days ships will be in Dubrovnik, because you don’t want to be there on those days.

How many days do you want to spend in Dubrovnik and Split? Did you want do day trips or spend time in island in lieu of time in Split and Dubrovnik? You could theoretically spend one night each in Split and Dubrovnik and three nights on one island, two on another. Or, more time in Split and Dubrovnik and one island.
No need for a car. See the catamaran schedules on the Jadrolinija and Krilo websites to get a sense of the routes and schedules. There are some other providers too (as well as bus travel, in the case of Korcula), but those are the main two and probably the most convenient.
It's hard to beat Hvar and Korcula, but the less mentioned ones like Lastovo are gorgeous, too.