Assuming, for the sake of argument, that certain killing of dogs for fur is done in a way that involves a material amount of pain, would this then be relatively strong support for the view that such killing would be immoral?

On that assumption (including the assumption that the pain is of the morally relevant kind - let me call it 'suffering'), the immoral thing would be inflicting that suffering, not killing the animal nor wearing its fur.

Suppose a certain amount of suffering was practically unavoidable in killing that type of animal. Then, strictly speaking, what was morally relevant would not actually be the suffering but the difference in suffering between that necessitated by the humane killing and that which would be experienced if the animal eventually died of natural causes.

Your assumption then leads to some conclusions that I don't think most animal rights supporters would like. For instance, not only would it be immoral to breed any such animals in the first place (because of the inevitable suffering that they would experience during their lives and deaths), but the only moral environmental policy would be to reduce the number of wild animals capable of suffering to the absolute minimum needed for humans to thrive.