In some ways, the political/medical alignment problem turns Cialdini on his head in that once one sees other people as "rabid GOPers" then an appeal using Cialdini's "Social Proof" would turn the target of persuasion against one's subject of persuasion.
If I, an unexpert, were to work on the problem, I would use the ambiguity of risk to hold the vax door open (if the target did other vaccines, did not hold strong religious whatevers against vax, etc). With the door open I might take advantage of information asymmetries and use Cialdini's "scarcity," letting them know that I know a particular place has vaccine X today which is better than vaccine Y because of some reasons, and that vaccine X is in short supply so if they were going to do it, this is a great opportunity.
For example, if you have someone who is total avaxx, how should you make him feel in order to change his mind?