Jadrolinija and www.krilo.hr/en/ operate the coastal catamaran routes from Dubrovnik to Split (and vice versa, of course) although neither have a sailing exactly on the route that you want.
With Kapetan Luka you can definitely do Dubrovik - Mljet - Korcula. (Note: you can also do the Dubrovnik - Mljet portion with G&V Line in case that helps - just be aware of the different towns on Mljet the catamarans come in to!
From Korcula, you'd then need to travel on to Hvar Town on Hvar - there's no direct sailing to Stari Grad. Equally, there are no sailings from Stari Grad to the island of Brac. So you'd either have to bus it from Hvar Town to Stari Grad and then back to Hvar Town to get to Milna or Bol on Brac (or you could also get to Jelsa for a different catamaran sailing to Bol).
From any of those places on Brac, it's pretty straightforward to get to Split.