If the dog had been given a different name or had not been run over, none of this would be an issue.
However, we are where we are....
It's not the "N-word" name of the dog that is offensive, it is the fact that the same word has been used millions of times to abuse people somehow perceived as of lesser value.
Most of us will feel some faint embarrasment whenever this anachronistic subject is raised as we can see all sides of the argument. However, there are those who either wish to use the dog's name as a bizarre and slightly sinister rallying cry for all that is perceived to be British, and there are also those who wish to rewrite history and probably exhume the dog as well.
I prefer the majority position, and surely some civilised solution can be developed, perhaps by rehanging the painting and by omitting the dog's name. We are supposed to be polite and considerate of the feelings of others, after all.