Does Greene advocate opposition against 'Capital Punishment' through the "Case for the Defence".
In my opinion, Greene does not advocate against capital punishment in this story. In fact, I do not think that this really has much to do with capital punishment.
I suppose you could argue that Greene is saying that juries are too likely to convict the wrong person. You could say that by mixing up the two men, Mrs. Salmon was showing us that it is too easy to get the wrong person convicted. In that case, it is an argument against the death penalty -- it is saying that eyewitness testimony is unreliable.
By contrast, you might say that the story is advocating capital punishment and vengeance. You might say this because of the fact that the crowd pushes one of the men under the bus. You can argue that Greene is saying that it is important to take revenge for crimes.
However, I do not think that the story is really about that. The circumstances are too unique -- the idea of identical twins. This makes it more of a trick ending than a real argument for or against capital punishment.