Start the new game.

[Dr.Vaфе!ь]

I have Soro dlc (tm) character, do I need to start a new character to fully benefit from three bros dlc?
I recall if you had a character without dlc and in process acquire said sorobor dlc the soroborean trader still won't travel with you to sorobor area.

[Aegix Drakan]

The Three Brothers DLC mostly happens after a base game questline is over. So no, you don't need to start over.