Don't know your (kid's) habits and your interests but I would suggest visiting Zagreb. I think it is ideal time to visit Zagreb during August (unitl Aug15th most locals are on holidays) and the city is half-empty, but everything is open, so great to explore. My suggestion would be 2 nights in DBV, then take a plane to ZAG (5 daily during August, RT ~100-130 euros at the moment, buy early because this flights are sold out during summer, often only 6AM departure available). Spend 2-3 nights in ZAG. Then take a plane back to SPU (buy multi-city so it will cost as RT to DBV on Croatia Airlines - OU), spend 2-3 nights in Split. To get back to DBV (I suggest that you have the final night in DBV) take a boat (catamaran) to DBV. There is 1 daily catamaran line Split - Dubvrovnik, via Brač,Hvar,Korcula,Mljet islands. So consider this as stopover if you have some days to spare and you are willing to have additional stopover.

If you are interested in swimming and lounging on beaches, then I would go +1 with JBord itinerary, islands are known for good beaches.