You would need to give up a feat for this, give up AOOs and you would do very little extra damage on your turn.
At 4th level comparing ASI to CBE:
Dex ASI: 1d4+1d6+4Dex+2Dueling = 12 DPR
CBE: 2d6+6Dex = 13DPR
You have a 5% greater chance to hit, which will increase damage by a little over half a point per turn beyond this.
When you count this against lost AOOs and turns where you use a different bonus action, I think you will actually be behind.
The character I am talking about is going to be a Assassin-Conquest Paladin multiclass so he will have numerous bonus actions available between cunning action, steady aim, Paladin spells and Hex.