Rent a car or hire a driver from Split to Zagreb?

There's nothing about Croatia that would make me think you need a lot of hand-holding, to pull this off, so I don't see why you shouldn't do this yourself. I haven't driven in Europe at all, much less in Croatia, so I don't know what driving would be like. There's bus service between Split and Zagreb taking as little a 5 hours or a bit less (according to the schedule, anyway). That link goes to Rome2Rio.com, whose schedule/duration/fare information is not to be trusted. But it identifies the bus companies running that route and gives links to their websites. Rail service is slower than the best buses. I've taken quite a few Croatian buses; they were comfortable. There is usually a small charge for placing a bag in the hold under the bus; if you encounter this, the driver isn't trying to scam you.

If your tour doesn't cover the Plitvice Lakes National Park, that would be my #1 recommendation. There are a couple of direct buses from Split to the park, apparently taking about 4-1/2 hours (if Rome2Rio is accurate here). At least prior to the pandemic, service to Zagreb was considerably more frequent. A good visit to the park really calls for spending one night there so you can avoid the terribly crowding caused by day-trippers.

Zagreb is a nice city and worth a day (or at least two if you like art--there are lots of art museums).

With four of you and the cost level in Croatia (somewhat lower than in France, italy, etc.), I think a private driver to Plitvice (and I guess a separate driver on to Zagreb the next day) might be affordable.

Where do you plan to travel after?
North or south? Or east

I'd suggest you establish the basics first. Croatia is pretty big. You'd be hard pressed, for example, to start your 2-3 days in Dubrovnik and expect to see Zagreb or Plitvice, too. What is the itinerary for your "coastal tour"? Are your interests nature vs. museums, or a mix of both? Art vs. history, or both? Do you have a budget? What is the proposed itinerary (and perhaps cost) for the three-day extension of the tour vs. planning and doing it yourself? Are you comfortable planning and going it alone, or do you prefer having a guide and/or the ease of having someone else plan to follow-on three days?

Getting around Croatia is pretty easy, both by public transportation and driving. Do you prefer one over the other? Or will it depend on your itinerary, or perhaps the costs of each? Once you zero in on these issues, it will be easier to offer suggestions.

I've driven rental cars on 2 trips in Croatia - along the coast from Dubrovnik (rented the car as I was leaving town) up to Zadar, and from Zagreb (again, rented the car as I was leaving) to Plitvice and around much of Istria. Generally it's easy driving, with the exception of some town centers.

Unless you'd prefer to leave the driving to someone else, having a rental car gives you a flexibility that you might enjoy. Google Maps worked great.

Driving is easy in Croatia, especially since most locals won’t use the new roads because of the toll. you can pay with a credit card. But, if you want a driver/tour guide I have one I can recommend. We have hired him, and others who work for him, on 2 trips to Croatia, the last just this past September. we used him in Dubrovnik to Medjugorje (as a tour guide/driver) then Split to Zadar (as a driver). His name is Mate Perajica and his email address is mateperajica@gmail.com. He speaks, reads, and writes English and has a great sense of humor.

I recommend renting a car because that gives you so much more flexibility. We rented a car in September 2019. My husband does all the driving and I can tell you it was very easy driving. We picked up the rental car the day we left Dubrovnik, and drove all the way to Zagreb, stopping along the way in Split, Plitvice and then on to Zagreb. The drive from Split to Zagreb is very easy.

If you are flying out of Split you may want to stay local.

If you are flying out of Zagreb, or could, a fast bus to Zagreb with the Arriva bus line - see their website - is easy. Let them drive and pay the tolls and gas and insurance. Book four seats at once and it will be like your own car. Masks mandatory. The seat numbers are hidden low on the seat armrest somewhere, usually. As noted, they charge a small extra fee in cash ( kunas - 10 or 20) and give you a receipt for your bag under the bus - line up with others handing over bags , your seat is already reserved. Small packs can go by your feet or on the overhead shelf. Bring snacks, it only stops once or twice.

The Zagreb bus depot (Autobusni Kolodvor) has many willing cabbies ready to take you to your hotel. Approach the front of their line.

Not sure what museums will be open then in Zagreb old, Upper Town, but I just enjoyed my first climb up the lookout Tower with the clock gun. Old friends took me on the Funicular railway up to it when I admitted I always just climbed the stairs. Both memorable.
A short walking tour of old and new Zagreb with a local guide would be well worth it. A piece of wood leaning against a house across the sidewalk means they are repairing the roof or earthquake damage, so do not walk under it. Solid boards protect main doors.