General Discussion

As a general rule, anyone can license musical works from the performing rights organizations (ASCAP, BMI, SESAC, etc.). All you have to do is pay the appropriate licensing fee and you get to use it. All of these public objections are just for show - but legally meaningless.

The only potential out is that some of the performing rights organizations have specific licensing for political purposes. If the artist licenses their works through one which has a separate political use license, the artist can purportedly exclude their works from political use. But the validity of a license which excludes some, but not all, works from political use (especially if it is a party-specific use has not yet been tested.

If nothing else - making a public statement or even sending a cease and desist letter is NOT a valid way to opt out of permitting your works to be used for political purposes.