On the one hand, this is a scary reminder that our copyright laws are way too strict. On the other hand, I'm willing to bet that this copyright claim is on behalf of some Obama fans in the music industry who want to make trouble for Romney. (And maybe it's a little bit of the music industry hitting back against Republicans in general, who steal music all the time.) I don't think the copyright claim would hold up in court—this has to be considered fair use—but BMG Rights Management could sure make it difficult for Romney's lawyers.
The primary question would be: Does the song directly relate to the content of the ad? Well, not really. Kinda. Since Romney's people are saying Obama is in bed with donors, his Al Green impersonation would be considered wooing, I guess. I think it's more likely that the Romney team will just drop the ad; it's not very effective and it continues the narrative for the Obama campaign's more effective ad that inspired this one.