Uh, no. What "PR battle" is Rush "winning," when the only folks likely to hear the challenge are those who already listen to his show?
Fine entertainment, I suppose, but politically it's just noise that will be ignored by everybody who's not already tuned in.
Uh no, he wins by preventing the other side from pursuing whatever false charge they have made against him, and making them look stupid in the process. He’s done this with every phoney controversy the other side stirs up.