Let him go.
The White Legs get massacred at Great Salt Lake by the 80s which is a far worse fate than having a bullet in your head. Not to mention, it teaches both Joshua and the Sorrows a good lesson. Mercy is an option, and they both need to know that.
But you can't know that will happen. Mercy is an option, but it shouldn't be available for someone such as Salt-Upon-Wounds.
I let him live. I already won. Letting him leave with his tail between his legs is a fitting punishment. Either his leadership will suffer, or his own tribe would blame the defeat on him and kill him. Or in the unlikely case, his tribe learns to leave others alone. (doesn't happen)
Everything after the defeat is for them to decide. The Sorrows and the Dead Horses have learned to fight and survive on their own. And Joshua learns about forgiveness or something.
And before anyone says, "Why let him leave? What if he comes back stronger than before?" The White Legs can't come back stronger than they once were.
They suffered immense casualties at the hand of The Courier and Joshua Graham. The tribes of Zion are now experienced at combat. Their leader's spirit has been broken, which is why Salt starts begging The Courier to reason with Graham to let him live.
And The White Legs can only survive by raiding. They can't farm. There is no evidence that they get food by hunting like nomadic tribes. They don't even have a place to call home. Face it. The White Legs are too weak to fight any other tribes and steal their food. They have been defeated. The only reason to kill Salt-Upon-Wounds is for revenge.
That's the way I see it.
Again, you can't know that the White Legs won't reform and come back. Not in their original strength, but as you said the only one they can sustain themselves is via raiding. They will probably get destroyed even if you let Salt go (I've never seen the ending), but the point is that it sends a clear message to the White Legs and there is more of a chance that they will cause damage should he flee than if he is killed. How will allowing their leader (who has committed atrocities all throughout Zion) to walk free break their will anymore than him being executed?
And you could call it vengeance, but men have to be accountable for their actions. Salt doesn't get to walk away after what he's done IMO. Revenge, justice, I don't see the distinction in his case.