We are doing exactly the same holiday in the opposite direction, Dubrovnik to Split. We have small kids in tow and will spend a week self catering in Milini, a week in a static Caravan on Hvar and one week in Supetar also self catering. We will not hire a car, so will be taking water taxis and ferries.
This will be our third time in Croatia, the previous times we stayed in Supetar on Brac, so I can only vouch for Supetar. It is very child friendly, more so than Bol (in my opinion), and there is a lovely sandy beach to the west of the town. And for a day trip you can take the ferry to Split.