The campaign ad has been removed from Carnahan's web site and YouTube, so unfortunately I can't independently evaluate it. (Update: here it is, at least for now.) But the complaint alleges that it was a "smear ad" that falsely implies that Fox and Wallace endorsed Carnahan's campaign. The complaint says that the 32-second ad uses "an essentially verbatim copy of a 30-second clip of both video footage and voice-over commentary appropriated from" an interview Wallace conducted with Blunt in 2006. The complaint also seems to say that the Fox footage included in the Carnahan ad included only Wallace's questions -- but not Blunt's answers:
The defendant’s conduct in stealing only certain footage from the [Fox] Interview is also false and misleading: Wallace’s tough questions were included, but Blunt’s answers and explanations were not.
The Kansas City Star described the ad as follows:
Without having seen the ad myself, I'm not going to offer an opinion on the ultimate merits of the suit. But I do have a bit of skepticism about the copyright claim, for at least two reasons. First, the complaint repeatedly emphasizes the alleged reputational damage to Fox for use of the footage. Even assuming that the ad does falsely imply that Fox and/or Wallace are endorsing Democrat Carnahan (a dubious proposition, it seems to me), reputational damage is just not a cognizable copyright interest. And second, the complaint asserts that the campaign's use of the Fox footage "allows Defendant to profit commercially without paying the traditional price." But that statement appears to contradict the thrust of the complaint, which is that Fox would never license such footage to a campaign, because it would damage its reputation. In other words, there is no "price" here, "traditional" or otherwise. Moreover, courts have rejected the argument that campaign uses of third-party material are "commercial" simply because they are used to solicit contributions. See American Family Life Insurance Co. v. Hagan, 266 F. Supp. 2d 682, 697 (N.D. Ohio 2002) (use of trademark in a political campaign ad was “properly classified not as a commercial transaction at all, but completely noncommercial, political speech”); MasterCard International Inc. v. Nader 2000 Primary Committee, Inc., 2004 WL 434404 (S.D.N.Y. Mar. 8, 2004) (even if a candidate’s ad resulted in increased contributions, the ad would still not be “commercial;” “If so . . . all political campaign speech would also be ‘commercial speech’ since all political candidates collect contributions”).
As for the right of publicity claims, I am not familiar enough with the specifics of Missouri law to say anything too definite. I would just point out that courts are very protective of First Amendment interests in the political context, see, e.g., Meyer v. Grant, 486 US 414 (1988) (First Amendment interests are "at its zenith" in the political realm), and there are strong arguments for limiting right of publicity claims to truly commercial uses of an individual's name and likeness.
(Updated with additional detail about the ad.)