Most car rental companies let you drop the car off after hours - I have done that numerous times in Croatia and elsewhere. Just check with the individual car company. Sometimes you drop the keys in a drop box; sometimes you leave them at a hotel front desk nearby. Always take pictures of the car where you have dropped it off as well, to verify no damage or anything.
Try EconomyCarRentals and Auto Europe for decent prices - or Sixt (www.sixt.com).
Whether you'd want a car after Split depends on what you wish to do afterward. If you are just going directly to Dubrovnik, it will probably save you a few hours just to drive there and drop the car. Having a car in Dubrovnik itself is kind of a pain - expensive to park it or forcing you to stay away from the old town where parking is easier. A car could be handy for day trips away from Dubrovnik though, but you could do that by bus or by taking a tour.
DO KNOW that driving from Split to Dubrovnik actually requires driving through a small slice of Bosnia ("Neum corridor") as you drive south - look on a Google map. That means that, technically, you must cross a border and drive out of the EU into Bosnia (and then right back into Croatia soon after), and that may cost more or require a green card (insurance card). Some car companies make exceptions for driving just through this small slice of Bosnia - check with them.
With a car longer you could also stop somewhere between Split and Dubrovnik - a detour to Mostar, BiH or maybe stop at an island like Korcula (which you could also do by ferry/catamaran). Depends how much time you have and what you are interested in doing.