I recommend reaching out to Ivan Ćosić from MC-Travel Croatia on Facebook. He offers private car transfers and is based in Split, though he covers all of Croatia. I've worked with him before and had great experiences. Reliable, friendly, and knows the best local spots.
Zagreb to Split with a stop at either Plitvice Lakes or Krka National Park is a popular transfer option he can easily arrange.
For Split to Dubrovnik, he typically charges around €400, which includes a stop in Ston (famous for oysters and the walls) and other points of interest if you'd like to explore a bit along the way.
Definitely worth checking with him for an exact quote and options.