A perfectly reasonable thing for Barbour to do would be to say "I can't make them take the picture down. However, I wish they would as it wrongfully implies that I endorse their views, which I don't."
But, that wouldn't be a prudent thing to do, politically. So he won't.
Haley Barbour Flashback (from the 10/20/82 NYT):
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But the racial sensitivity at Barbour headquarters was suggested by an exchange between the candidate and an aide who complained that there would be ''coons'' at a campaign stop at the state fair. Embarrassed that a reporter heard this, Mr. Barbour warned that if the aide persisted in racist remarks, he would be reincarnated as a watermelon and placed at the mercy of blacks.